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Introduction to Random Walks on Noncommutative

Spaces

Philippe Biane

To cite this version:

Philippe Biane. Introduction to Random Walks on Noncommutative Spaces. Uwe Franz,Michael Schurmann. Quantum Potential Theory, Springer, pp.61-116, 2008, Lecture Notes inMathematics, <10.1007/978-3-540-69365-9 3>. <hal-00786127>

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Introduction to Random Walkson Noncommutative Spaces

Philippe Biane

Abstract We introduce several examples of random walks on noncommuta-tive spaces and study some of their probabilistic properties. We emphasizeconnections between classical potential theory and group representations.

1 Introduction

Whereas random walks form one of the most investigated class of stochasticprocesses, their noncommutative analogues have been studied only recently.In these lectures I will present some results on random walks which take theirvalues in noncommutative spaces. The notion of a noncommutative space hasemerged progressively from the development of quantum physics, see e.g. [C].The key idea is to consider not the space itself but the set of real, or complexfunctions on it. For a usual space, this forms an algebra, which is commu-tative by nature. A noncommutative space is given by a noncommutative(usually complex) algebra which is to be thought of as the algebra of com-plex functions on the space. We shall explain this idea in more details insection 2, and in particular define noncommutative probability spaces. Oncenoncommutative spaces have been defined in this way it is easy to define ran-dom variables, and stochastic processes taking their values in these spaces.Rather than starting an abstract theory, these lectures will consist mainly ina collection of examples, which I think show that this notion is interestingand worth studying. We shall begin with the most simple stochastic processnamely the Bernoulli random walk. We shall show how to quantize it inorder to construct the quantum Bernoulli random walk. Simple as it is this

P. BianeInstitut d’electronique et d’informatique Gaspard-Monge, 77454 Marne-la-Vallee Cedex 2,Francee-mail: [email protected]

U. Franz, M. Schurmann (eds.) Quantum Potential Theory. 61

Lecture Notes in Mathematics 1954.

c© Springer-Verlag Berlin Heidelberg 2008

62 P. Biane

noncommutative stochastic process exhibits quite deep properties, related togroup representation and potential theory. Actually interpreting it as a ran-dom walk with values in a noncommutative space, the dual of SU(2), we willbe lead to define random walks with values in duals of compact groups. Thestudy of such random walks in the case of special unitary groups uncoversconnections with potential theory, in particular with the Martin boundary.We will investigate more on these connections. From the Bernoulli randomwalk we can take limit objects, as in the central limit theorem. One of theseobjects is a noncommutative Brownian motion which we construct as a fam-ily of operators on a Fock space and interpret then as a continuous timestochastic process with independent increments, with values in the dual ofthe Heisenberg group. We then extend this construction to more generalnoncompact locally compact groups. Finally we will also start to considerquantum groups in the last section.

The next section consists in preliminaries about C∗ and von Neumannalgebras and noncommutative spaces.

2 Noncommutative Spaces and Random Variables

2.1 What are Noncommutative Spaces?

The random walks that we are going to study take their values in noncom-mutative spaces, so we should start by making this notion more precise. Inmany parts of mathematics, one studies spaces through the set of functionsdefined on them. There can be many kind of functions, e.g. measurable, inte-grable, continuous, bounded, differentiable, and so on. Each property of thefunctions reflects a property of the space on which they are defined. Some-times, in probability theory for example, one is even not interested at all inthe space, but only in the functions themselves, the random variables. Alsovery often the set of complex functions considered determines completely theunderlying space. This is the case for example for compact topological spaces,determined by their algebra of continuous functions, or differentiable mani-folds which are determined by their smooth functions. The common featureshared by these situations is that all these spaces of complex valued func-tions are commutative algebras. It has been realized, since the beginning ofquantum mechanics that one can obtain a better description of nature byrelaxing this commutativity hypothesis. Henceforth we shall consider a noncommutative space as given by a complex algebra, which plays the role ofspace of functions on the space. We will see many examples throughout theselectures. Some algebras may come equipped with a supplementary struc-ture, for example an antilinear involution, a norm, a preferred linear form,a topology, etc... Actually most of the times these algebras will be algebras

Introduction to Random Walks on Noncommutative Spaces 63

of operators on some complex Hilbert space H , and the involution will begiven by the adjoint operation. We shall decribe the kind of algebras we willconsider, mainly C∗ and von Neumann algebras. We will use the languageand some basic results in the theory of these objects, but we will need nodeep knowledge of them. We will only assume that the reader is familiar withthe spectral theorem for selfadjoint operators on a Hilbert space. We referfor example to the treatises [D1], [D2] or [T] for more details.

2.2 C∗ Algebras

A C∗ algebra is a normed ∗-algebra which is isometric with a subalgebra ofthe algebra B(H) of all bounded operators on some complex Hilbert space H ,stable under taking the adjoint, and closed for the operator norm topology.Elements in a C∗-algebra of the form aa∗ for some a ∈ A are called positive.Positive elements are exactly the selfadjoint positive operators which belongto the algebra.

Let X be a locally compact topological space, then the algebra of complexcontinuous functions on X , vanishing at infinity, is a C∗-algebra, and the fa-mous Gelfand-Naimark theorem states that any commutative C∗-algebra isisomorphic to such an algebra. The topological space is compact if and only ifthe algebra has a unit, and there is a one to one correspondence between thepoints of the space and the characters of the algebra, that is, the continuousalgebra homomorphisms with values in the complex numbers, or equivalentlywith the maximal closed ideals, therefore the space is unique up to home-omorphism and can be recovered from the algebra. It is usually denotedby spec(A) if A is the C∗-algebra. Thus we should think of a C∗-algebraas providing the algebra of continuous functions on some noncommutativespace. Note that C∗ algebras are closed under continuous functional calcu-lus, namely if a is a self-adjoint element in a C∗ algebra, and f a continuousfunctions on its spectrum, then the operator f(a) also belongs to the algebraC. This can be easily seen by approximating uniformly f by polynomials onthe spectrum of a.

If A ⊂ B(H) is a C∗-algebra, then the multiplier algebra M(A) of A is theset of all operators x such that xA ⊂ A and Ax ⊂ A. It is a C∗ algebra witha unit, containing A. It coincides with A if and only if A has a unit. If A isabelian, then M(A) is just the algebra of all bounded continuous functions onspec(A). In the noncommutative situation, it corresponds to the Stone-Cechcompactification of the topological space underlying the algebra.

Continuous positive linear functionals on a C∗ algebra play the role ofpositive bounded measures. Here positivity for a functional means that itis positive on positive elements. Again in the commutative case, by Riesz’theorem, such linear functionals correspond to finite positive Borel measureson the underlying topological space. Positive linear functionals of norm one

64 P. Biane

are called states, and correspond to probability measures in the commutativecase. A large supply of states is given by unit vectors in the Hilbert space onwhich the C∗ algebra acts. Indeed any such vector ψ defines a state by theformula

ωψ(a) = 〈aψ, ψ〉 a ∈ A

Given a self adjoint element a in a C∗ algebra A, and a state σ on A, thereexists a unique measure on R, with compact support, such that

σ(f(a)) =

∫

f(x)dμ(x) for all continuous f on R.

The support of this measure is included in the spectrum of a. The GNSconstruction assigns to every C∗ algebra, with a continuous positive linearfunctional σ, a representation of the algebra on a Hilbert space. A linearfunctional is called tracial if for any a, b ∈ A one has τ(ab) = τ(ba).

Each continuous map between topological spaces f : X → Y gives riseto a continuous algebra morphism Φf : C0(Y ) → C0(X); h �→ h ◦ f , andconversely any such algebra morphism comes from a continuous map, there-fore one can think of a homomorphism between C∗ algebras as a continuousmap between the underlying noncommutative spaces (with the direction ofthe arrows reversed). One must note however that there may exist very fewmorphisms between two C∗ algebras. For example there does not exist anynonzero homomorphism from the finite dimensional C∗ algebra Mn(C) toMm(C) if n > m. Indeed this is a purely algebraic fact, since Mn(C) isa simple algebra, if a homomorphism from Mn(C) is not injective, then itmust be 0.

2.3 von Neumann Algebras

Let S be a subset of B(H), then its commutant S′ is the set of boundedoperators which commute with every element of S. A von Neumann algebrais a subalgebra of B(H) which is closed under under taking the adjoint,and is equal to its bicommutant, i.e. the commutant of its commutant. Bythe von Neumann bicommutant theorem the von Neumann algebras are the∗-subalgebra of B(H), containing the identity operator, and closed for thestrong topology. Since the strong topology is weaker than the operator normtopology any von Neumann algebra is also a unital C∗ algebra, althoughgenerally too large to be interesting as such.

A von Neumann algebra is closed under Borel functional calculus, namelyif a ∈ M is self-adjoint and f is a bounded Borel function on the spectrum ofa, then f(a) belongs to M , and again, the same is true for f(a1, . . . , an) wherea1, . . . , an are commuting selfadjoint operators in the von Neumann algebra,and f is a bounded Borel function defined on the product of their spectra.

Introduction to Random Walks on Noncommutative Spaces 65

The nuance between C∗ algebras and von Neumann algebras can be graspedby looking at the commutative case. Indeed commutative von Neumann al-gebras correspond to measure spaces, more precisely any commutative vonNeumann algebra is isomorphic to the algebra L∞(X, m) where X is a mea-sure space and m a positive measure (the algebra actually depends only onthe class of the measure). This statement can be seen as a reformulationof the spectral theorem for commuting self-adjoint operators on a Hilbertspace. Therefore it is natural to think of von Neumann algebras as “algebrasof noncommutative random variables”. A normal state on a von Neumannalgebra is a positive linear form which is continuous for the σ-weak topologyand takes the value 1 on the unit. It corresponds, in the commutative case,to a probability measure, which is absolutely continuous with respect to themeasure m. We shall sometimes call a von Neumann algebra, with a normalstate, a “non commutative probability space”.

A weight on a von Neumann algebra is a map ϕ from the cone of posi-tive elements of the von Neumann algebra to [0, +∞], which is additive, andhomogeneous, i.e. ϕ(λx) = λϕ(x) for x positive and real λ > 0. A weight iscalled normal if supi∈I ϕ(xi) = ϕ(sup(xi)i∈I) for every bounded increasingnet (xi)i∈I . Coming back to the commutative case, weights are positive, pos-sibly unbounded measures, in the measure class of m. A weight μ is calledfinite if μ(1) < ∞, in this case μ is a multiple of a state.

Given a selfadjoint element, a ∈ M and a normal state σ on M , we denoteby μa the distribution of a, namely the measure such that

σ(f(a)) =

∫

f(x)dμa(x)

for all bounded Borel functions on spec(a). More generally if a1, . . . , an is afamily of commuting self-adjoint operators in M , their joint distribution isthe unique probability measure μa1,...,an on Rn such that

σ(f(a1, . . . , an)) =

∫

f(x)dμa1,...,an(x)

for all bounded Borel function f on Rn.Let N ⊂ M be a von Neumann subalgebra, and σ a state on M , then a

conditional expectation of M onto N is a norm one projection σ(.|N) suchthat σ(a|N) = a for all a ∈ N , σ(abc|N) = aσ(b|N)c for all a, c ∈ N, b ∈ M ,and σ(σ(b|N)) = σ(b) for all b ∈ M . Given M, N and σ, such a map neednot exist, but it always exists, and is unique, if σ is tracial.

We will consider spatial tensor products of von Neumann algebras. IfA ⊂ B(H) and B ⊂ B(K) are two von Neumann algebras, their algebraictensor product acts on the Hilbert space H ⊗K, and the spatial tensor prod-uct of A and B is defined as the von Neumann algebra generated by thistensor product. Given an infinite family of von Neumann algebras (Ai; i ∈ I)equipped with normal states ωi it is possible to construct an infinite tensor

66 P. Biane

product ⊗i(Ai, ωi) which is a von Neumann algebra with a state ⊗iωi. Oneconsiders operators of the form ⊗i∈Iai where ai ∈ Ai and ai �= Id only fora finite number of i ∈ I. These generate an algebra, which is the algebraictensor product of the Ai. One can define a positive linear functional on thisalgebraic tensor product by ω(⊗i∈Iai) =

∏

ωi(ai). The GNS constructionthen yields a Hilbert space H , with a pure state on B(H), and the von Neu-mann algebra tensor product is the von Neumann algebra in B(H) generatedby this algebraic tensor product.

2.4 Random Variables, Stochastic Processes

with Values in some Noncommutative Space

Given a von Neumann algebra M equipped with a normal state σ, and a C∗

algebra C, a random variable with values in C (or, more appropriately, in thenoncommutative space underlying C), is a norm continuous morphism fromC to M . The distribution of the random variable ϕ : C → M is the stateon C given by σ ◦ϕ. If the algebra C is commutative, then it corresponds tosome topological space, and the state σ ◦ ϕ to a probability measure on thisspace. When C = C0(R) there exists a self-adjoint element a ∈ M such thatϕ(f) = f(a) for all f ∈ C, and we are back to the situation in the precedingsection, where the distribution of a was defined. We will call the state σ ◦ ϕthe distribution of the random variable ϕ. More generally a family of randomvariables with values in some noncommutative space, indexed by some set, isa stochastic process.

If A and B are two C∗-algebras, a positive map Φ : A → B is a linearmap such that Φ(a) is positive for each positive a ∈ A. When A and B arecommutative, thus A = C0(X) and B = C0(Y ), such a linear map can berealized as a measure kernel k(y, dx) where for each y ∈ Y one has a finitepositive measure k(y, dx) on X . If A and B are unital, and Φ(I) = I then thiskernel is a Markov kernel, i.e. all measures are probability measures. Thuswe see that the generalization of a Markov kernel to the non-commutativecontext can be given by the notion of positive maps. It turns out howeverthat this notion is slightly too general to be useful and it is necessary torestrict oneself to a particular class called completely positive maps.

Definition 2.1. linear map between two C∗ algebras A and B is called com-pletely positive if, for all n ≥ 0, the map Φ ⊗ Id : A ⊗ Mn(C) → B ⊗ Mn(C)is positive. It is called unit preserving if furthermore Φ(Id) = Id.

We shall consider semigroups of unit preserving, completely positive maps ona C∗ algebra C. These will be indexed by a set of times which will be eitherthe nonnegative integers (discrete times) or the positive real line (continuoustime). Thus a discrete time semigroup of unit preserving, completely positivemaps on a C∗ algebra C will be a family (Φn : C → C)n≥0 of completely

Introduction to Random Walks on Noncommutative Spaces 67

positive maps, such that Φn ◦ Φm = Φn+m. In continuous time we will havea family (Φt)t∈R+ which satisfies Φt ◦Φs = Φt+s. In the discrete time settingone has Φn = (Φ1)

n and the semigroup is deduced from the value at time 1.We shall denote generally the time set by T when we do not specify whetherwe are in discrete or continuous time.

Definition 2.2. Let C be a C∗ algebra, then a dilation of a semigroup(Φt)t∈T of completely positive maps on C is given by a von Neumann algebraM , with a normal state ω, an increasing family of von Neumann subalgebrasMt; t ∈ T , with conditional expectations ω(.|Mt), and a family of morphismsjt : C → (M, ω) such that for any t ∈ T and a ∈ C, one has jt(a) ∈ Mt andfor all s < t

ω(jt(a)|Ms) = js(Φt−s(a)) (2.1)

A dilation of a completely positive semigroup is the analogue in noncommu-tative probability of a Markov process, and the equation (2.1) expresses theMarkov property of the process: the conditional expectation of the future onthe past is a function of the present.

Given a completely positive semigroup and an initial state, a dilation al-ways exists [S].

Consider a completely positive semigroup on a C∗ algebra C, and letB ⊂ C be a commutative C∗ subalgebra, thus isomorphic to C0(X) forsome locally compact topological space X . If the image algebras jt(B); t ∈ Tgenerate a commutative von Neumann algebra N ⊂ M , then there exists aprobability space (Ω,F , P ) such that (N , ω) ∼ L∞(Ω,F , P ), and randomvariables Xt : Ω → X , corresponding to the morphisms jt restricted toB, which form a classical stochastic process. If furthermore the C∗ algebraB is invariant by the completely positive semigroup, then this semigroupdefines a Markov semigroup of transition probabilities on the space X , andthe stochastic process (Xt)t∈T is a Markov process with these probabilitytransitions. This remark will be at the basis of many constructions of classicalstochastic processes starting from quantum ones.

Once a dilation of a completely positive semigroup is given, one can com-pute, for times t1 < · · · < tn, and a1, . . . , an ∈ C,

ω(jt1(a1) . . . jtn(an)) = σ(Φt1 (a1(Φt2−t1(a2(. . . Φtn−tn−1(an)) . . .)

where σ = ω ◦ j0 is the initial state on C (the distribution of the processat time 0). Observe however that when the algebras jt(C); t ∈ T do notcommute, this condition does not specifiy the values of

ω(jt1(a1) . . . jtn(an))

when the times t1, . . . , tn are not ordered. We will say that two dilationsj(1), ω1 and j(2), ω2, are equivalent if one has

ω1(j(1)t1 (a1) . . . j

(1)tn

(an)) = ω2(j(2)t1 (a1) . . . j

(2)tn

(an))

68 P. Biane

where t1, . . . , tn is an arbitrary sequence of times (i.e. not necessarily in-creasing) and a1, . . . , an ∈ C. Thus if C is a commutative algebra, then allcommutative dilations with the same initial distribution are equivalent, butfor a given semigroup of completely positive maps there may exist a lot ofnon equivalent dilations. Actually we shall encounter in these lectures somenatural noncommutative dilations of Markov semigroups on classical spaces!An important source of such dilations comes from restrictions: if B ⊂ C isa subalgebra and the completely positive semigroup leaves B invariant, thenthe restriction of (jt)t∈T to the subalgebra B is a dilation of the restrictionof the completely positive semigroup.

3 Quantum Bernoulli Random Walks

3.1 Quantization of the Bernoulli Random Walk

Our first example of a quantum random walk will be the quantization ofthe simple (or Bernoulli) random walk. This is just the random walk whoseindependent increments have values ±1. In order to quantize it we will replacethe set of increments {±1} by its quantum analogue, namely the space oftwo by two complex matrices, with its structure of C∗-algebra. The subsetof hermitian operators is a four dimensional real subspace, generated by theidentity matrix I as well as the three matrices

σx =

(

0 11 0

)

σy =

(

0 −ii 0

)

σz =

(

1 00 −1

)

The matrices σx, σy , σz are the Pauli matrices. They satisfy the commutationrelations

[σx, σy ] = 2iσz; [σy , σz] = 2iσx; [σz , σx] = 2iσy. (3.1)

The group SU(2) acts by the automorphisms A → UAU∗ on this C∗-algebra.We observe that this group is much larger than the group of symmetries of thetwo points space (which consists just of a two elements group). This actionleaves the space generated by I invariant, and acts by rotations on the realthree dimensional space generated by the Pauli matrices. Indeed the innerproduct on the space of hermitian matrices 〈A, B〉 = Tr(AB) is invariant byunitary conjugation.

A state ω on M2(C) is given by a positive hermitian matrix S with trace1, by the formula

ω(A) = Tr(AS).

The most general such matrix can be written as

S =1

2

(

1 + u v + iwv − iw 1 − u

)

Introduction to Random Walks on Noncommutative Spaces 69

where (u, v, w) ∈ R3 satisfies u2 + v2 + w2 ≤ 1. The extreme points on theunit sphere (sometimes called the “Bloch sphere” in the physics litterature),correspond to pure states, when S is a projection on a one dimensional sub-space. Any hermitian operator has a two-point spectrum, hence in a stateω its distribution is a probability measure on R supported by at most twopoints. In particular, for each of the Pauli matrices, its distribution in thestate ω is a probability measure on {±1}, given by

P (σx = 1) =1 + v

2P (σy = 1) =

1 + w

2P (σz = 1) =

1 + u

2(3.2)

Mimicking the construction of a random walk, we use the infinite productalgebra (M2(C), ω)⊗N (recall the construction of section 2.3). For each Paulimatrix we build the matrices

xn = I⊗(n−1)⊗σx⊗I⊗∞ yn = I⊗(n−1)⊗σy⊗I⊗∞, zn = I⊗(n−1)⊗σz⊗I⊗∞

which represent the increments of the process. It is easy to see that, forexample, the operators xn, for n ≥ 1, form a commuting family of operators,which is distributed, in the state ω∞, as a sequence of independent Bernoullirandom variables.

Then we put

Xn =

n∑

i=1

xi; Yn =

n∑

i=1

yi Zn =

n∑

i=1

zi.

This gives us three families of operators (Xn)n≥1; (Yn)n≥1 and (Zn)n≥1 onthis space.

We observe that each of these three families consists in commuting op-erators, hence has a joint distribution. It is not difficult to check that thisdistribution is that of a Bernoulli random walk, whose increments have dis-tribution given by the probability distributions on {±1} of formula (3.2).The three families of operators, however do not commute. In fact using thecommutation relations (3.1) one sees that for n, m positive integers,

[xn, ym] = 2izmδnm

and[Xn, Ym] = 2iZn∧m (3.3)

as well as the similar relations obtained by cyclic permutation of X, Y, Z.We will call the family of triples of operators (Xn, Yn, Zn); n ≥ 1 a quantumBernoulli random walk. We shall later interpret this noncommutative processas a random walk with values in some noncommutative space, however for themoment we will study some of its properties related to the automorphismsof the algebra M2(C).

70 P. Biane

3.2 The Spin Process

Because of the rotation invariance of the commutation relations (3.1), we seethat for any unitary matrix U we can define conjugated variables

xUn = I⊗(n−1) ⊗ UσxU∗ ⊗ I⊗∞

yUn = I⊗(n−1) ⊗ UσyU∗ ⊗ I⊗∞

zUn = I⊗(n−1) ⊗ UσzU

∗ ⊗ I⊗∞

and

XUn =

n∑

i=1

xUi ; Y U

n =

n∑

i=1

yUi ZU

n =

n∑

i=1

zUi

then this new stochastic process is obtained from the original quantumBernoulli random walk by a rotation matrix. It follows by a simple com-putation, using the commutation relations, that X2

n + Y 2n + Z2

n is invariantunder conjugation, namely one has

X2n + Y 2

n + Z2n = (XU

n )2 + (Y Un )2 + (ZU

n )2

for any unitary matrix U .

Lemma 3.1. For all m, n ≥ 1 one has

[X2n + Y 2

n + Z2n, X2

m + Y 2m + Z2

m] = 0

Actually we shall prove that [Xn, X2m + Y 2

m + Z2m] = 0 if m ≤ n. This follows

from the computation

[Xn, Y 2m] = [Xn, Ym]Ym + Ym[Xn, Ym] = 2i(ZmYm + YmZm)

[Xn, Z2m] = [Xn, Zm]Zm + Zm[Xn, Ym] = −2i(ZmYm + YmZm)

Using invariance of the commutation relations by cyclic permutation ofX, Y, Z we also have [Yn, X2

m + Y 2m + Z2

m] = [Zn, X2m + Y 2

m + Z2m] = 0, and

the result follows. ⊓⊔We deduce from this that the family of operators (X2

n + Y 2n + Z2

n); n ≥ 1 iscommutative, and therefore defines a classical process.We shall compute its dis-tribution now. For this we introduce the operators Sn =

√

X2n + Y 2

n + Z2n + I.

By the preceding Lemma, the family (Sn)n≥0 is a commuting family of op-erators, therefore one can consider their joint distribution.

Theorem 3.2. Let us take for ω the tracial state, then the operators (Sn;n≥ 1) form a Markov chain, with values in the positive integers, such thatP (S1 = 2) = 1, and transition probabilities

p(k, k + 1) =k + 1

2k; p(k, k − 1) =

k − 1

2k.

Introduction to Random Walks on Noncommutative Spaces 71

In order to prove the theorem, it is enough to consider the process up to afinite time n. So we shall restrict considerations to a finite product M2(C)⊗n

acting on (C2)⊗n. We remark that the state ω⊗n on (C2)⊗n is the uniquetracial state, and it gives, to every orthogonal projection on a subspace V , the

value dim(V )2n . We shall find a basis of this space consisting of joint eigenvectors

of the operators (Sk; 1 ≤ k ≤ n). For this we analyze the action of theoperators A+

n := 12 (Xn − iYn) and A−

n = 12 (Xn + iYn). One has

A+n =

n∑

j=1

I⊗(j−1) ⊗ α+ ⊗ I⊗n−j A−n =

n∑

j=1

I⊗(j−1) ⊗ α− ⊗ I⊗n−j

where

α+ =

(

0 01 0

)

α− =

(

0 10 0

)

.

Let us call e0, e1 the canonical basis of C2. An orthonormal basis of (C2)⊗n

is given by the vectors eU ; U ⊂ {1, 2, . . . , n} where eU = ei1 ⊗ . . .⊗ein , ik = 1if k ∈ U and ik = 0 if k /∈ U . In terms of this basis the action of A+

n and A−n

is given byA+

n eU =∑

k/∈U eU∪{k}A−

n eU =∑

k∈U eU\{k}ZneU = (n − 2|U |)eU

Let us consider the vector e∅ = e⊗n0 and its images by the powers of A−

n ,normalized to have norm one. These are the vectors

εjn =

√

j!(n − j)!

n!

∑

U⊂{1,2,...,n};|U|=j

eU j = 0, 1, . . . , n

and these vectors are orthogonal. The action of the operators A+n , A−

n , Zn onthese vectors is given by

A+n εj

n =√

(j + 1)(n − j)εj+1n

A−n εj

n =√

j(n − j + 1)εj−1n

Znεjn = (n − 2j)εj

n

Snεjn = (n + 1)εj

n

In particular we see that these vectors belong to the eigenspace of Sn ofeigenvalue n + 1. We shall generalize this computation to find the commoneigenspaces of the operators S1, S2, . . . , Sn.

Lemma 3.3. Let J = (j1, . . . , jn) be a sequence of integers such that

• i) j1 = 2• ii) ji ≥ 1 for all i ≤ n• iii) |ji+1 − ji| = 1 for all i ≤ n − 1

72 P. Biane

then there exists a subspace HJ of (C2)⊗n of dimension jn = l + 1, whichis an eigenspace of S1, . . . , Sn, with respective eigenvalues j1, . . . , jn, and anorthonormal basis (φ0, . . . , φl) such that

A+n φj =

√

(j + 1)(l − j)φj+1

A−n φj =

√

j(l − j + 1)φj−1

Znφj = (l − 2j)φj

Furthermore, the spaces HJ are orthogonal and (C2)⊗n = ⊕JHJ .

Proof of the lemma. We shall use induction on n. The lemma is truefor n = 1, using φ0 = e0; φ1 = e1. Assume the lemma holds for n, andlet J = (j1, . . . , jn) be a sequence satisfying the conditions i), ii), iii), ofthe lemma. We shall decompose the space HJ ⊗ C2 as a direct sum of twosubspaces. Let the vectors ψj and ηj be defined by

ψj =

√

l − j + 1

l + 1φj ⊗ e0 +

√

j

l + 1φj−1 ⊗ e1 j = 0, . . . , l + 1

and

ηj =

√

j + 1

l + 1φj+1 ⊗ e0 −

√

l − j

l + 1φj ⊗ e1 j = 0, . . . , l − 1

It is easy to check that these vectors form an orthonormal basis of the tensorproduct HJ ⊗ C2, and a simple computation using

Xn+1 = Xn + I⊗n ⊗ σx, Yn+1 = Yn + I⊗n ⊗ σy , Zn+1 = Zn + I⊗n ⊗ σz

shows that these vectors have the right behaviour under these operators. ⊓⊔We can now prove theorem 3.2, indeed for any sequence satisfying the

hypotheses of lemma 3.3 one has

P (S1 = j1, . . . , Sn = jn) =jn

2n=

j12

j22j1

. . .jn

2jn−1

where the right hand side is given by the distribution of the Markov chain oftheorem 3.2. ⊓⊔

We shall give another, more conceptual, proof of the former result insection 5, using the representation theory of the group SU(2) and Clebsch-Gordan formulas. For this we shall interpret the quantum Bernoulli randomwalk as a Markov process with values in a noncommutative space, but weneed first to give some definitions pertaining to bialgebras and group alge-bras, which we do in the next section.

Introduction to Random Walks on Noncommutative Spaces 73

4 Bialgebras and Group Algebras

4.1 Coproducts

Let X be a finite set, and F(X) be the algebra of complex functions on X .A composition law on X is a map X × X → X . This gives rise to a unitpreserving algebra morphism ∆ : F(X) → F(X × X), where F(X × X)is the algebra of functions on X × X , and one has a natural isomorphismF(X × X) ∼ F(X) ⊗ F(X). Conversely, such an algebra morphism ∆ :F(X) → F(X)⊗F(X) comes from a composition law, and many propertiesof the composition law can be read on it. For example, associativity translatesinto coassociativity of the coproduct which means that

(∆ ⊗ I) ◦ ∆ = (I ⊗ ∆) ◦ ∆

whereas commutativity gives cocommutativity for the coproduct, whichmeans that

v ◦ ∆ = ∆

where v : F(X) ⊗ F(X) → F(X) ⊗ F(X) is the flip automorphism v(a ⊗b) = b ⊗ a. In order to obtain an analogue of a composition law in thenoncommutative context, one can define a coproduct for any algebra A as amorphism ∆ : A → A⊗A, however in general the algebraic tensor product istoo small for obtaining interesting examples. Think for example to the caseA = Cb(X), X a locally compact space, and see that A ⊗ A ⊂ Cb(X × X) isa small subspace. We shall therefore consider coproducts which take valuesin a suitable completion of the algebraic tensor product. An algebra endowedwith a coassociative coproduct is called a bialgebra. If the algebra is a C∗

(resp. a von Neumann) algebra and the tensor product is the minimal C∗

product (resp. the von Neumann algebra tensor product), then one has aC∗ (resp. a von Neumann) bialgebra.

Some further properties of a coproduct are the existence of the dual notionof the unit element and the inverse, which are respectively called a counit,ε : A → C and an antipode, i : ∆ → ∆.

A Hopf algebra is a bialgebra with a unit and an antipode, satisfying somecompatibility conditions. I refer for example to [K] for an exposition of Hopfalgebras and quantum groups.

4.2 Some Algebras Associated to a Compact Group

We shall investigate in more details the notions above in the case of the groupalgebra of a compact group, which we assume for simplicity to be separable.Recall that every representation of a compact group can be made unitary, andcan be reduced to an orthogonal direct sum of irreducible representations. The

74 P. Biane

right regular representation of a compact group G is the representation of Gon L2(G, m) (where m is a Haar measure on G) by right translations ρgf(h) =f(hg−1) and the left regular representation acts by left translations λgf(h) =f(gh). A fundamental theorem is the Peter-Weyl theorem. It states that everyirreducible representation of G arises in the decomposition of the left (orright) regular representation, actually one has an orthogonal direct sum

L2(G) = ⊕χ∈GEχ

where G is the (countable) set of equivalence classes of irreducible representa-tions of G, and for χ ∈ G, Eχ is the space of coefficients of the representationi.e. the vector space generated by functions on G of the form

f(g) = 〈χ(g)u, v〉

where u, v are vectors in the representation space of χ (〈., .〉 being an invarianthermitian product on the space). The space Eχ is finite dimensional, itsdimension being dim(χ)2, and it is an algebra for the convolution producton G, isomorphic to the matrix algebra Mn(C) with n = dim(χ). We shalldenote this space Mχ when we want to emphasize its algebra structure.

We shall describe several algebras associated to G. The first one is theconvolution algebra A0(G) generated by the coefficients of the finite di-mensional representations. As a vector space it is the algebraic direct sumA0(G) = ⊕χMχ ∼ ⊕χMdim(χ)(C). There are larger algebras such as L1(G)the space of integrable functions (with respect to the Haar measure on G),and C∗(G) the C∗ algebra generated by L1(G). This algebra consists of se-quences (mχ; χ ∈ G) such that mχ ∈ Mχ; |mχ| → 0 as χ → ∞.

The multiplier algebra of C∗(G) coincides with the von Neumann algebraA(G), which is generated (topologically) by the left translation operatorsλg; g ∈ G, it consists of sequences (mχ; χ ∈ G) such that supχ |mχ| < ∞. Inboth cases, the norm in these algebras is supχ |mχ|. Note that the left andright translation operators λg and ρg are unitary, and the right translationoperators generate the commutant of A(G).

When the group G is abelian, there is a natural isomorphism, given byFourier transform, between the algebra A(G) and L∞(G) where here G isthe group of characters of G (this is consistent with our earlier notationsince then all irreducible representations of G are one dimensional and arethus characters). This group of characters is a discrete abelian group, and itsHaar measure is the counting measure.

In the general case, the algebra L∞(G) is isomorphic with the center ofA(G), since a bounded function on G can be identified with a sequence(mχ)χ∈G of scalar operators.

Any closed subgroup of G generates a von Neumann subalgebra A(H) fur-thermore the coproduct ∆ restricts to this subalgebra and defines a coproduct∆ : A(H) → A(H) ⊗ A(H). If the subgroup is abelian, then this subalgebrais commutative and is isomorphic to the algebra L∞(H),

Introduction to Random Walks on Noncommutative Spaces 75

Finally we shall also use the algebra A(G) =∏

χ Mχ which consists in

unbounded operators on L2(G), with common dense domain ⊕χEχ (algebraicdirect sum), affiliated with the von Neumann algebra A(G). One has naturalinclusions

A0(G) ⊂ C∗(G) ⊂ M(C∗(G)) = A(G) ⊂ A(G)

The algebra A(G) ⊗ A(G) is an algebra of operators on the algebraic directsum ⊕χ,χ′Eχ ⊗Eχ′ and we denote by A(G)⊗A(G) its completion for simpleconvergence on ⊕χ,χ′Eχ ⊗ Eχ′ . One has

A(G) ⊗ A(G) =∏

χ

Mχ ⊗∏

χ

Mχ ⊂ A(G)⊗A(G) ∼∏

χ,χ′

Mχ ⊗ Mχ′

The ∗ algebra structure extends obviously to A(G), and an element is positiveif and only if its components are positive.

4.3 The Coproduct

The coproduct formula ∆ : λg → λg ⊗λg extends by linearity and continuityto the von Neumann algebra A(G) if we use the von Neumann algebra tensorproduct. It defines a structure of cocommutative von Neumann bialgebra onA(G). One can also define an extension of the coproduct

∆ : A(G) → A(G)⊗A(G).

Indeed it is easy to check that for any χ ∈ G, and a ∈ Mχ, the operator ∆(a)is nonzero on the space Eχ′ ⊗ Eχ′′ if and only if χ has a non zero multiplic-ity in the decomposition of the tensor product representation Eχ′ ⊗ Eχ′′ . Itfollows that for any sequence (aχ)χ∈G ∈ ∏

χ∈G Mχ the sum∑

χ∈G ∆(aχ) isa finite sum in each component Eχ ⊗ Eχ′ therefore it defines an element in

A(G)⊗A(G) ∼ ∏

χ,χ′ Mχ ⊗ Mχ′ .One can define the convolution of two finite weights μ and ν by the formula

μ ∗ ν = (μ ⊗ ν) ◦ ∆

since the coproduct is cocommutative and coassociative, one checks that thisis an associative and commutative operation.

4.4 The Case of Lie Groups

For compact Lie groups there is another algebra of interest, which is theenvelopping algebra of the Lie algebra. Recall that the Lie algebra of a Lie

76 P. Biane

group is composed of right invariant vector fields on the group. As such itacts on the L2 space as a family of unbounded operators. Therefore the Liealgebra is a subspace of the “big” algebra A(G), and the algebra generatedby this subspace is naturally isomorphic to the envelopping Lie algebra (forsimply connected groups). The elements of the Lie algebra have the followingbehaviour with respect to the extended coproduct on A(G)

∆(X) = X ⊗ I + I ⊗ X

This can be seen by taking derivatives with respect to s in the equation

∆(esX) = esX ⊗ esX

where esX ; s ∈ R is the one parameter subgroup generated by X .

4.5 States and Weights

A normal state ν on A(G) is determined by its value on the generators λg ,thus by the function φν(g) = ν(λg). It is a classical result that a function φon G corresponds to a state on C∗(G), and to a normal state on A(G), if andonly if it is a continuous positive definite function on the group, satisfyingφ(e) = 1.

Every normal weight on A(G) is given by a sequence of weights (νχ)χ∈G

on each of the subalgebras Mχ, therefore for a weight ν on A(G) there existsa sequence of positive elements fχ ∈ Mχ such that νχ(a) = Tr(afχ) forall a ∈ Mχ. Conversely any such sequence fχ defines a normal weight, thus

normal weights on A(G) correspond to positive elements in A(G).

5 Random Walk on the Dual of SU(2)

5.1 The Dual of SU(2) as a Noncommutative Space

We shall now interpret the process constructed in section 3 as a randomwalk on a noncommutative space. For this we consider the group SU(2) ofunitary 2×2 matrices with determinant 1. It is well known that the irreduciblerepresentations of this group are finite dimensional, moreover, for each integern ≥ 1 there exists, up to equivalence, exactly one irreducible representationof dimension n, therefore one has

A(SU(2)) = ⊕∞n=1Mn(C)

Introduction to Random Walks on Noncommutative Spaces 77

as a von Neumann algebra direct sum and similarly the C∗ algebra of SU(2)can be identified with the algebra of sequences (Mn)n≥1 where Mn is ann × n matrix and one has ‖Mn‖ → 0 as n → ∞. We shall interpret thisC∗-algebra as the space of “functions” vanishing at infinity on some non-commutative space. The associated von Neumann algebra corresponds to thespace of “bounded functions” on our noncommutative space. In order to get abetter picture of this space it is desirable to have a geometric understandingof its structure. For this we first note that this space has a continuous groupof symmetries, since the group SU(2) acts on the algebra by inner automor-phisms. Since the elements ±I act trivially, this is really an action of thequotient SU(2)/{±I} which is isomorphic with the group SO(3), thereforethis space has a three dimensional rotational symmetry. We can understandthis symmetry by looking at some special elements in the larger algebraA(SU(2)), which correspond to “unbounded functions”. Let us consider theself-adjoint elements corresponding to the Pauli matrices, viewed as elementsof the Lie algebra of SU(2) (or rather the complexified Lie algebra). Thesedefine unbounded operators on L2(SU(2)), which we shall denote X, Y, Z,and which lie in A(SU(2)). A good way to think about these three functionsis as three “coordinates” on our space, corresponding to three orthogonaldirections. Each of these elements has a spectrum which is exactly the set ofintegers (which you can view as the group dual to the one parameter groupgenerated by one of these Lie algebra elements). Moreover this is true also ofany linear combination xX +yY +zZ with x2 +y2 +z2 = 1. This means thatif you are in this space and try to measure your position, you can measure,as in quantum mechanics, one coordinate in some direction (x, y, z) using theoperator xX + yY + zZ, and you will always find an integer. Thus the spacehas some discrete feature, in that you always get integer numbers for yourcoordinates, but also a continuous rotational symmetry which comes from theaction of SU(2) by automorphisms of the algebra. This is obviously impos-sible to obtain in a classical space. Of course since the operators in differentdirections do not commute, you cannot measure your position in differentdirections of space simultaneously. What you can do nevertheless is measuresimultaneoulsy one coordinate in space, and your distance to the origin. Thislast measurement is done using the operator D =

√I + X2 + Y 2 + Z2 − I

which is in the center of the algebra A(SU(2)), and therefore can be measuredsimultaneously with any other operator. Its eigenvalues are the nonnegativeintegers 0, 1, 2 . . . , and its spectral projections are the identity elements ofthe algebras Mn(C), more precisely, one has in A(SU(2))

D =

∞∑

n=1

(n − 1)IMn(C)

We thus see that the subalgebra Mn(C) is a kind of “noncommutativesphere of radius n − 1”, and moreover by looking at the eigenvalues ofthe operators xX + yY + zZ in the corresponding representation, we see

78 P. Biane

A(SU(2))

Fig. 1 Noncommutative space underlying A(SU(2)).

that on any “radius” of this sphere, corresponding to a direction of space,the coordinate on this radius can only take the n values n − 1, n−3,n− 5, . . . ,−n + 1.

If we rescale the noncommutative sphere of large radius to have radius1, it looks more and more like a classical sphere, see e.g. [Rie] for a precisestatement.

5.2 Construction of the Random Walk

Let ω be a state on M2(C), which we can also consider as a state onA(SU(2)) by the projection A(SU(2)) → M2(C). Let us consider the in-finite tensor product algebra, with respect to the product state ν = ω⊗∞ onN = ⊗∞

1 A(SU(2))).Let T : N → N be defined by ∆ ⊗ s where s : A(SU(2))[2,∞[ →

A(SU(2))[3,∞[ is the obvious shift isomorphism. Let jn : A(SU(2)) → Nbe the morphisms defined by by jn = T n ◦ i where i(a) = a ⊗ I∞ is theGNS representation of A(SU(2)) associated with ω, acting on the first fac-tor. Note that one has actually N = ⊗∞

1 M2(C) and i(a) = ρ2(a)⊗ I∞ whereρ2 is a two dimensional irreducible representation. The morphisms jn can beextended to the large algebra A(SU(2)) and for V in the Lie algebra, usingthe formula for the coproduct one checks that

jn(V ) =

n∑

i=1

I⊗(i−1) ⊗ V ⊗ I⊗∞

Introduction to Random Walks on Noncommutative Spaces 79

Thus the quantum Bernoulli random walk is obtained by taking the operators(jn(X), jn(Y ), jn(Z)) where X, Y, Z are the Lie algebra elements correspond-ing to the Pauli matrices. For n ≥ 0, we let Nn be the algebra generated bythe first n factors in the tensor product. There exists a conditional expecta-tion ν(.|Nn) with respect to the state ν onto such a subalgebra, it is givensimply by I⊗ω∞ on the factorization N = A(SU(2))⊗n⊗A(SU(2))⊗[n+1,∞[.

The family of morphisms (jn)n≥1 form a noncommutative process wihvalues in the dual of SU(2). The following proposition is left to the reader,as an exercise in manipulation of coproducts.

Proposition 5.1. The family of morphisms jn; n ≥ 1, together with the fam-ily of algebras N ,Nn; n ≥ 1, and the conditional expectations ν(.|Nn), forma dilation of the completely positive map A(SU(2)) → A(SU(2)) given byΦω = (I ⊗ ω) ◦ ∆.

Let us now consider the three one-parameter subgroups generated by thePauli matrices, they consist respectively of the matrices

(

cos θ i sin θi sin θ cos θ

) (

cos θ sin θ− sin θ cos θ

) (

eiθ 00 e−iθ

)

θ ∈ [0, 2π[

Each of these subgroups generates a commutative von Neumann subalgebraof A(SU(2)), which is isomorphic with the group von Neumann algebra ofthe group U(1) of complex numbers of modulus 1. Such a von Neumann alge-bra is isomorphic, by Pontryagin duality, to the algebra of bounded functionson the dual group, therefore the restriction of the dilation to this subalgebraprovides a random walk on this dual group, which is isomorphic to Z. Thuswe recover, from the abstract considerations on duals of compact groups, ourconcrete Bernoulli random walks. In terms of our noncommutative space, wecan observe a particle undergoing this random walk along any fixed direc-tion of space, and what we see is a Bernoulli random walk (recall that thecoordinate in some fixed direction of space can only take integer values).

We now turn to the spin process which can be interpreted in terms of therestriction of the dilation (jn)n≥1 to the center of the group algebra A(G).

This center consists of operators of the form (mχ; χ ∈ G) where each mχ isa scalar operator in Mχ, it is a commutative algebra, isomorphic with the

algebra of bounded functions on G (recall that we have assumed that G iscountable). Equivalently it is the algebra generated by the operator D i.e.the algebra of operators of the form f(D) where f is a bounded complexfunction on the nonnegative integers. It is also easy to compute the restric-tion of the completely positive map Φω to this center. It is given by theClebsch-Gordan formula which computes the decomposition into irreducibleof a tensor product of representations of the group SU(2). What we need isthe formula (where ρn is the n-dimensional irreducible representation)

ρ2 ⊗ ρn = ρn−1 ⊕ ρn+1

80 P. Biane

which tells us that the Markov chain can jump from n to either n−1 or n+1,furthermore the transition probabilities are proportional to the dimensionsof the targets, since we are in the trace state. We see that the restriction ofjn to this algebra thus corresponds to the spin process.

Finally there are other commutative algebras which are invariant underthe completely positive map. They are generated by the center of A(SU(2))and by a one parameter subgroup. Each such algebra is a maximal abeliansubalgebra of A(SU(2)), and its spectrum can be identified with the set ofpairs (m, n) of integers such that n ≥ 1 and m ∈ {n−1, n−3, . . .−n+1}. Onecan compute the associated Markov semigroup, using the Clebsch Gordanformulas for products of coefficient functions of the group SU(2). Howeverthe images of such an algebra by the morphisms (jn)n≥1 do not commute.Thus in this way we get a noncommutative dilation of a purely commutativesemigroup. We will come back to this Markov chain when we study Pitman’stheorem and quantum groups.

6 Random Walks on Duals of Compact Groups

It is easy to generalize the construction of the preceding section by replacingthe group SU(2) by an arbitrary compact group G. We will do a constructionparallel to the usual construction of a random walk on a group. Let φ0 and φbe continuous positive definite functions on G, with φ0(e) = φ(e) = 1, thusthese functions correspond to normal states ν0 and ν on A(G). The state ν0

will play the role of initial condition of our Markov chain, whereas the stateν represents the distribution of the increments. To the state ν is associateda completely positive map

Φν : A(G) → A(G) Φν = (I ⊗ ν) ◦ ∆.

The completely positive map generates a semigroup Φnν ; n ≥ 1. We now build

the infinite tensor product N = A(G)∞ with respect to the state ν0 ⊗ ν⊗∞,and obtain a noncommutative probability space (N , ω). Let T : N → Ndefined by ∆⊗s where s : A(G)[1,∞[ → A(G)[2,∞[ is the obvious isomorphism.Let jn : A(G) → N be the morphisms defined by induction jn = T n ◦ i wherei(a) = a⊗ I∞ is the inclusion of A(G) into the first factor (strictly speakingthis is an inclusion only if the state ν0 is faithful).

Let us translate the above construction in the case of an abelian group,in terms of the dual group G. The states ν and ν0 correspond to probabilitymeasures on G, the probability space is now the product of an infinite numberof copies of G, with the product probability ν0 ⊗ ν⊗∞, and the maps jn

correspond to functions Xn : G∞ → G given by Xn(g0, g1, . . . , gk, . . .) =g0g1 . . . gn−1. We thus recover the usual construction of a random walk.

Introduction to Random Walks on Noncommutative Spaces 81

For n ≥ 0, we let Nn be the algebra generated by the first n + 1 factorsin the tensor product. There exists a conditional expectation ω(.|Nn) withrespect to the state ω onto such a subalgebra, it is given simply by I ⊗ ν∞

on the factorization N = A(G)⊗(n+1) ⊗A(G)⊗∞.

Proposition 6.1. The morphisms (jn)n≥0, together with the von Neumannalgebras N ,Nn, n ≥ 0 and the state ω, form a dilation of the completelypositive semigroup (Φn

ν )n≥1, with initial distribution ν0.

The proof of this proposition follows exactly the case of SU(2) treated in thepreceding section.

Let H be a closed commutative subgroup of G, then its dual group His a countable discrete abelian group. The von Neumann subalgebra A(H)generated by H in A(G) is isomorphic to L∞(H), and the restriction ofthe positive definite function φ to H is the Fourier transform of a probabilitymeasure μ on H . The coproduct ∆ restricts to a coproduct on the subalgebragenerated by H , thus the images of A(H) by the morphisms jn generatecommuting subalgebras of N . These restrictions thus give a random walk onthe dual group H , whose independent increments are distributed accordingto μ.

We now consider another commutative algebra, namely the center Z(G) ofA(G). Recall that this center is isomorphic with the space L∞(G) of boundedfunctions on the set of equivalence classes of irreducible representations of G.As a von Neumann algebra of operators on L2(G), it is generated by theconvolution operators by integrable central functions (recall that a functionf on G is central if it satisfies f(ghg−1) = f(h) for all h, g ∈ G).

Proposition 6.2. The algebras jn(Z(G)); n ≥ 1 commute.

Proof. Let a, b ∈ Z(G) and let k ≤ l we have to prove that jk(a) and jl(b)commute. Let a′ = i(a), b′ = i(b), and note that a′ and b′ belong to the centerof N . One has

jk(a)jl(b) = T k(a′)T k(T l−k(b′))= T k(a′T l−k(b′))= T k(T l−k(b′)a′)= T l(b′)T k(a′)= jl(b)jk(a)

⊓⊔Furthermore one has.

Proposition 6.3. If the function φ is central then Φν(Z(G)) ⊂ Z(G).

Proof. Indeed if ψ is central function, belonging to A0(G), then Φν(ψ) = φψis a central function on the group, and thus defines an element of Z(G). ⊓⊔

We deduce from the preceding propositions that, in the case when the in-crements correspond to a central state, the restriction of the dilation (jn)n≥0

to the center of the algebra A(G) defines a Markov process on G, whose

82 P. Biane

transition operator is given by the restriction of Φν to the center Z(G). Weshall now give a more concrete form of the transition probabilites of thisMarkov chain.

Proposition 6.4. For χ an irreducible character of G, let

φχ =∑

χ′∈G

hφ(χ′, χ)χ′

be the expansion of the positive definite central function φχ into a combinationcharacters, then the probability transitions of the Markov chain obtained fromthe restriction of (jn)n≥1 to the center are given by

pφ(χ′, χ) =dim(χ)

dim(χ′)hφ(χ′, χ)

Proof. For χ ∈ G, the convolution operator associated with the functiondim(χ)χ is the minimal projection of Z(G) associated with χ. In other wordsit corresponds to the indicator function 1χ in the isomorphism of L∞(G)with Z(G). The transition operator for the restriction of the dilation to thecenter is the restriction to this center of Φω. On the other hand, the transitionprobabilities pφ(χ, χ′) are related to these indicator functions by Φω(1χ′) =∑

χ pφ(χ′, χ)1χ. The conclusion follows immediately. ⊓⊔

7 The Case of SU(n)

7.1 Some Facts About the Group SU(n)

We shall investigate the quantum random walk defined in the precedingsection when the group G is the group SU(n) of unitary matrices with de-terminant 1. First we recall some basic facts about this group and its repre-sentations. Let T ⊂ SU(n) be the subgroup of diagonal matrices, which is amaximal torus. The group of characters of T is an n− 1 dimensional lattice,generated by the elements (the notation e is here to suggest the exponentialfunction)

e(ei)(

⎛

⎜

⎝

u1 0. . .

0 un

⎞

⎟

⎠) = uj

These elements satisfy the relation (written in additive notation) e1 +e2 + . . . + en = 0, which corresponds to the relation e(e1)e(e2) . . . e(en) = 1for the characters. We denote by P this group, and by P+ the subset ofpositive weights, i.e.

P+ = {m1e1 + . . . + mn−1en−1|m1 ≥ m2 ≥ . . . ≥ mn−1}

Introduction to Random Walks on Noncommutative Spaces 83

Fig. 2

We shall also need the set

P++ = {m1e1 + . . . + mn−1en−1|m1 > m2 > . . . > mn−1}

and note that the two sets are in bijection by

P++ = P+ + ρ (7.1)

where ρ = (n− 1)e1 + (n− 2)e2 + . . .+ en−1 is the half sum of positive roots.The symmetric group acts on this character group by permutation of the ei.

Below is a picture of P and P+ for the group SU(3). Thus P consists ofthe points in a triangular lattice in the plane, and P+ is the intersection ofthis triangular lattice with a cone, fundamental domain for the action of thesymmetric group S3. The subset P++ consists in points of P+ which are inthe interior of the cone.

Recall from the representation theory of the group SU(n) (see e.g. [BtD],[GW], or [Z]) that the equivalence classes of irreducible representations ofSU(n) are in one to one correspondence with the elements of P+, which arecalled “highest weights”. For each x ∈ P let e(x) be the associated characterof T, then the character of the representation with highest weight x ∈ P+

given, for u ∈ T, by Weyl’s character formula

χx(u) =

∑

σ∈Snǫ(σ)e(σ(x + ρ))(u)

∑

σ∈Snǫ(σ)e(σ(ρ))(u)

(7.2)

In particular the defining representation of SU(n) has character e(e1)+ . . .+e(en) corresponding to the highest weight e1. The normalized positive definite

84 P. Biane

1/3

1/3

1/3

Fig. 3 The random walk on the dual of the maximal torus for SU(3).

function on SU(n) corresponding to this character is φ(g) = 1nTr(g). We shall

investigate the quantum random walk associated with this positive definitefunction.

7.2 Two Classical Markov Chains

We shall obtain two classical Markov chains by restricting the Markov chainassociated with φ to suitable subalgebras of C∗(SU(n)). The first subalgebrais that generated by the maximal torus T. This algebra is isomorphic to thealgebra of functions vanishing at infinity on the dual group P. It is easy tosee that the restriction to the torus of the quantum random walk (jn)n≥0,constructed using φ, is a random walk on the lattice P with incrementsdistributed as 1

n (δe1 + . . . + δen). Its one step transition probabilities aregiven by

p1(x, y) = 1n if y ∈ {x + e1, . . . , x + en}

p1(x, y) = 0 if not

We give the picture for the case of the group SU(3).The second subalgebra is the center of C∗(SU(n)). By the preceding sec-

tion, it can be identified with the space of functions vanishing at infinity onP+. We shall rather use the identification of the set of irreducible represen-tations with P++ given by (7.1). As we saw in section 6 the restriction of thedilation (jn)n≥0 gives a classical Markov chain on the spectrum of the center,therefore we obtain in this way a Markov chain on P++, the generator of theMarkov chain being given by the restriction of the generator of the quantumMarkov chain on A(SU(2)).

Introduction to Random Walks on Noncommutative Spaces 85

We shall now derive a relation between these two Markov chains (the oneobtained from the maximal torus, and the one from the center). The set ofhighest weights, P++ is the intersection of P with the Weyl chamber, whichis the cone

C = {x1e1 + . . . + xn−1en−1|x1 > x2 > . . . > xn−1}

We consider now the random walk on P killed at the exit of this cone. Thusthe transition probabilites of this killed random walk are given by

p0(x, y) = 1n if y ∈ P++ ∩ {x + e1, . . . , x + en}

p0(x, y) = 0 if not.

The sum∑

y p0(x, y) is < 1 for points near the boundary, corresponding tothe fact that the random walk has a nonzero probability of being killed.

Let x ∈ P++ and consider the irreducible representation ξx of SU(n), withhighest weight x − ρ. We can use Weyl’s character formula (7.2) for decom-posing the representation ξx ⊗ ξn (where ξn is the defining representation ofSU(n)).

We remark that 1n

∑nj=1 e(ej) = 1

n!

∑

σ∈Sne(σ(e1)). One has

χnχx−ρ =1

n(e(e1) + . . . + e(en))

∑

σ∈Snǫ(σ)e(σ(x))

∑

σ∈Snǫ(σ)e(σ(ρ))

=

∑

τ∈Sn

∑

σ∈Snǫ(σ)e(σ(x) + τ(e1))(u)

n!∑

σ∈Snǫ(σ)e(σ(ρ))

=1

n

∑nj=1

∑

σ∈Snǫ(σ)e(σ(x + ej))

∑

σ∈Snǫ(σ)e(σ(ρ))

=1

n

∑

y∈{x+e1,x+e2,...,x+en}∩P++

χy−ρ

since if y = x + ej belongs to P+ \ P++ then it is fixed by some reflexion inSn and thus the sum

∑

σ∈Snǫ(σ)e(σ(x + ej)) vanishes.

For x ∈ P++, let us denote h(x) the dimension of the representationwith highest weight x− ρ. We conclude from the preceding computation andProposition 6.4, that the transition probabilities for the Markov chain onP++ are

q(x, y) =h(y)

h(x)p0(x, y) x, y,∈ P++ (7.3)

Since the transition operator is unit preserving, it follows in particular thatthe function h is a positive harmonic function with respect to the transitionkernel p0, i.e. satisfies

86 P. Biane

x

a

b

c

h(a)/3h(x)

h(b)/3h(x)

h(c)/3h(x)

Fig. 4

h(x) =∑

y

h(y)p0(x, y) for all x ∈ P++,

Again we draw the picture in the case of SU(3).Let us recall that, given transition (sub)probabilities p(x, y) for a Markov

chain, and a positive harmonic function h for p, i.e. a function such that

h(x) =∑

y

h(y)p(x, y) for all x

one calls the Markov chain with transition probabilities h(y)h(x)p(x, y) the

Doob conditioning, or h-transform, of the Markov chain with transitionprobabilities p.

We can summarize the preceding discussion in the following proposition.

Proposition 7.1. The Markov chain obtained by restriction to the centeris related to the random walk on the dual of the maximal torus by a killingat the exit of the Weyl chamber followed by a Doob conditioning using thedimension function on the set of highest weights.

For more information on Doob’s conditioning, I refer to the books byKemeny, Knapp and Snell [KKS], or Revuz [R].

One could ask whether this relation, between the Markov chains on thedual of the torus and on the center, holds for more general groups and positivedefinite functions. It turns out that the fundamental concept in this directionis that of a minuscule weight, see [B3] for more details.

Introduction to Random Walks on Noncommutative Spaces 87

8 Choquet-Deny Theorem for Duals of Compact Groups

8.1 The Choquet-Deny Theorem in an Abelian Group

As we have seen in the preceding section, the fact that the dimension functionis a positive harmonic function on the Weyl chamber plays an important rolein understanding the quantum random walk on the dual of SU(n). A naturalquestion arises, is this positive harmonic function unique? We shall answerthis question, which is purely a question of “classical” potential theory usingthe theory of quantum random walks. Actually we shall do this by extending awell known result of Choquet and Deny [CD] on harmonic measures for locallycompact abelian goups. The Choquet-Deny theorem describes completely thesolutions of the convolution equation

μ ∗ φ = φ (8.1)

on a locally compact abelian group G, where μ is a positive finite measure, andφ is an unknown positive measure. One assumes that the subgroup generatedby the support of μ is the whole group. In order to state the Choquet-Denytheorem, one needs to define the exponentials on the group G. These are thecontinuous functions f on G, with values in ]0, +∞[, which are multiplicative,i.e. satisfy

f(gh) = f(g)f(h)

for all g, h ∈ G. An exponential e : G →]0, +∞[ is called μ-harmonic if onehas

∫

Ge(−x)μ(dx) = 1. The set of μ-harmonic exponentials is a Borel subset

of the set of all continuous functions on G, which we denote by Eμ. Let φ bea positive measure on G of the form φ(dx) = e(x)dx where dx is the Haarmeasure on G and e is a μ-harmonic exponential, then one has for all positivemeasurable functions f

∫

G f(x)φ ∗ μ(dx) =∫

G

∫

G f(x + y)φ(dx)μ(dy)

=∫

G

∫

G f(x + y)e(x)dxμ(dy)

=∫

G

∫

G f(x)e(x − y)dxμ(dy)

=∫

Gf(x)e(x)dx

∫

Ge(−y)μ(dy)

=∫

Gf(x)φ(dx)

therefore the measure φ is μ-harmonic. i.e. satisfies the equation (8.1). TheChoquet-Deny theorem states that every solution is a convex combination ofsolutions of this kind.

Theorem 8.1 (Choquet-Deny). Assume that the subgroup generated bythe support of the measure μ is G, then every positive measure φ, solutionof the convolution equation (8.1), is absolutely continuous with respect ot theHaar measure on G, and its density has a unique representation as an integral

88 P. Biane

dφ(x)

dx=

∫

Eµ

e(x)dνφ(e)

where νφ is some finite positive measure on Eμ.

This result contains in particular the determination of all positive har-monic functions for the transition operator associated with the measure μ,indeed if h is such a positive μ-harmonic function, then the measure h(−x)dxis a μ-harmonic measure.

Note that a locally compact abelian group corresponds to a commutativeand cocommutative Hopf C∗-algebra. What we shall do next is to extend thistheorem to (a class of) cocommutative Hopf C∗-algebras. This will allow us toanswer the question about the uniqueness of the positive harmonic function.Before we state our analogue of the Choquet-Deny theorem on the dual of acompact group, we will first clarify some points about states and weights.

8.2 Some Further Properties of Duals of Compact

Groups

Let G be a compact group and A(G) be its von Neumann algebra. Thisvon Neumann algebra has a structure of cocommutative Hopf-von Neumannalgebra for the coproduct ∆, counit ε and antipode i given respectively bycontinuous linear extension of

∆(λg) = λg ⊗ λg ε(λg) = δe,g i(λg) = λg−1 .

We have seen in section 4.3 that the convolution of two finite normal weightsμ and ν on A(G) is defined by the formula μ ∗ ν = (μ ⊗ ν) ◦ ∆. Assumenow that μ is a positive finite weight and φ is positive, then we can write φas a sum of finite weights φ =

∑

χ φχ with respect to the restrictions of φto the subalgebras Mχ. The sum

∑

χ ν ∗ φχ is then a sum of positive finiteweights and thus it defines a (not necessarily normal) weight on A(G). Wecan therefore consider the equation (8.1) where this time μ and φ are positivenormal weights, with μ finite.

Next we define the notion of exponential. The following definition is astraighforward extension of the definition in the case of abelian groups.

Definition 8.2. A group-like element is a non zero element f of A(G), suchthat

∆(f) = f ⊗ f.

If this element is positive, then we call it an exponential.If μ is a finite weight, f is an exponential and μ(i(f)) = 1 then we call f

a μ-harmonic exponential.

Let f, g, h be positive elements in A(G) then one has the identity

tr(f ⊗ g∆(h)) = tr(i(h) ⊗ g∆(i(f))) (8.2)

Introduction to Random Walks on Noncommutative Spaces 89

which is easily checked on coefficient functions. Let g ∈ A(G) be associatedto μ, i.e. μ(.) = tr(g.), then it follows from 8.2 that for all h

φ ∗ μ(h) = tr(f ⊗ g∆(h))= tr(i(h) ⊗ g∆(i(f)))= tr(i(h)i(f))tr(gi(f))= φ(h)

Thus if f is a μ-harmonic exponential, and φ is the weight associated withf , then φ satisfies the equation (8.1).

8.3 The Analogue of the Choquet-Deny Theorem

We shall assume that the state ν satisfies a non degeneracy condition whichis the analogue, in the noncommutative setting, of the requirement that thesupport of the measure generates the whole group. This condition states thatfor every finite weight ρ on A(G) which is supported by some algebra Mχ for

χ ∈ G, there exists an integer n ≥ 1 and a constant c > 0 such that cν∗n ≥ ρ.

Theorem 8.3. Let φ be a ν-harmonic weight, then there exists a unique finitepositive measure on the set of ν-harmonic exponentials such that

φ =

∫

Eµ

e dmφ(e)

Sketch of proof. We let Sν be the convex cone of normal weights satisfying

φ ∗ ν ≤ φ

This cone is a closed subset of A, and we can use Choquet’s integral repre-sentation theorem to conclude that any element of this cone can be writtenas the barycenter of a measure supported on the set of its extremal rays. Nowone can analyze the extremal rays of this set, and see that such an extremalray consists in multiples of an exponential. The uniqueness argument comesfrom the existence of an algebra of functions on the cone, which separatespoints, see [B4] for details.

8.4 Examples

For a finite group, the set of exponentials is reduced to the identity.If the state ν is tracial, then there exists only one ν-harmonic exponential,

namely the identity. In the case of the group SU(n), the exponentials are inone to one correspondence with the positive elements in SL(n, C).

90 P. Biane

From these results we deduce that the positive harmonic function h ofProposition 7.1 was unique. Indeed in order to prove that a function h isthe unique (up to a multiplicative constant) positive harmonic function fora transition kernel p, it is enough to check that all the positive harmonic

functions for the relativized kernel h(y)h(x)p(x, y) are constant, which is what

the Choquet-Deny theorem tells us.

9 The Martin Compactification of the Dual of SU(2)

In the preceding section we have seen that the classical Choquet-Deny theo-rem about solutions of convolution equations has an analogue in duals of com-pact groups. This allows one to give an explicit description of all μ-harmonicpositive functions for a finite positive measure μ on a commutative group.By the Choquet-Deny theorem the positive μ-harmonic functions admit aunique integral representation in terms of minimal μ-harmonic functions,and these minimal harmonic functions can be identified with exponentials.The next natural question in this line of ideas is to describe the Martin com-pactification associated to a random walk with values in Zn. This Martincompactification provides a way to attach a boundary to the space Z

n inorder to obtain a compact space, where the boundary is naturally identifiedwith the set of minimal μ-harmonic functions. The Martin compactificationof Zn was computed by Ney and Spitzer in a classical paper [NS], wherethey showed that it consists in adding a sphere at infinity, and identifyingthis sphere with the set of minimal harmonic functions with the help of theGauss map. In this section we will describe the Martin compactification ofsome quantum random walks with values in the dual of SU(2). We will firstrecall some basic facts about classical Martin boundaries, then describe Neyand Spitzer’s theorem, before going to the case of the dual of SU(2).

9.1 The Martin Compactification for Markov Chains

Consider a Markov chain on a countable state space E. The Markov chainhas transition subprobabilities p(x, y), x, y ∈ E, i.e. we have

∑

y p(x, y) ≤ 1,so that the kernel is submarkovian and the process may die in a finite time.There is an associated transition operator given by

Pf(x) =∑

y∈E

p(x, y)f(y)

and the iterated operator is given by n-step transition probabilities

Pnf(x) =∑

E

pn(x, y)f(y)

Introduction to Random Walks on Noncommutative Spaces 91

We assume that the associated Markov chain is transient, so that the potential

U =∞∑

n=0

Pn

is finite, i.e. the function

u(x, y) =

∞∑

n=0

pn(x, y) < ∞

Let us choose an initial distribution r(dx) such that the function rU(y) =∑

E u(x, y)r(dx) is everywhere > 0. The Martin kernel is defined by

k(x, y) =u(x, y)

rU(y)

It follows from the Harnack inequalities that the functions k(x, .) form a uni-formly continuous family on E. The Martin compactification of the Markovchain is the smallest compact topological space Eu, which contains E as adense subset, and such that these functions extend continuously to the bound-ary ∂Eu = Eu \ E. This space exists because the functions k(x, .) separatethe points of E and because of the uniform continuity. For any ξ ∈ ∂Eu

the function x �→ k(x, ξ) is a p-harmonic function. Recall that a positive p-harmonic function f on E is called minimal if for every positive p-harmonicfunction g satisfying g ≤ Cf for some C > 0, one has actually g = cf forsome constant c. One can prove that any minimal p-harmonic function f ,which is r-integrable, is a multiple of k(., ξ) for some ξ ∈ ∂Eu. The subset∂Em of ξ ∈ ∂Eu such that k(., ξ) is minimal is a Borel subset, and one canprove that any positive p-harmonic function f , which is r-integrable, admitsa representation

f =

∫

∂Eu

k(., ξ)dmf (ξ)

with a unique positive measure mf .

9.2 The Martin Compactification of Zd

When the Markov chain is a random walk on Zd, with increments distributedas μ, a (sub)probability measure on Zd, we have seen that every positive μ-harmonic function admits an integral representation in terms of exponentials.When the increments of the random walk are integrable, Ney and Spitzer havedetermined explicitly the Martin compactification of Zd, which we shall nowdescribe. Let φ : R

d → [0, +∞] be the function

φ(x) =∑

y∈Zd

e〈x,y〉μ(dy)

92 P. Biane

We assume that it is finite in a neighbourhood of the set

Eμ = {x|φ(x) = 1}

Then the set Eμ is the boundary of the convex set {x|φ(x) ≤ 1} and it iseither reduced to a point or homeomorphic to a sphere. In the latter case thehomeomorphism can be expressed thanks to the Gauss map as

∇φ

‖∇φ‖Since φ is convex this is a homeomorphism from Eμ onto the unit sphere. Neyand Spitzer proved that the Martin compactification is homeomorphic to theusual compactification of Zd by a sphere at infinity, where the identificationbetween the sphere and the set of minimal μ-harmonic functions is providedby the map above.

9.3 Noncommutative Compactifications

Before we investigate the problem of finding an analogue of the Ney-Spitzertheorem for the dual of SU(2) let us translate in the noncommutative lan-guage the notion of a compactification of a topological space. So let X be atopological space, and X a compact space such that X ⊂ X is a dense opensubset, and ∂X = X \ X is the boundary, then C(X) is can be identifiedwith a subalgebra of Cb(X), the algebra of all bounded continuous functionson X , and one has an exact sequence

0 → C0(X) → C(X) → C(∂X) → 0

where the first map is the continuous extension, to X, by 0 on the boundary,of a function on X , and the second map is the restriction to a closed subset.A compactification of C0(X) is thus given by a certain commutative C∗-subalgebra of the multiplier algebra of C0(X), containing C0(X). In the caseof the Martin compactification this subalgebra is just the algebra generatedby C0(X) and by the functions y �→ k(x, y). The vector space generated byfunctions of the form y �→ k(x, y) can be identified with the image of theMartin kernel, considered as an integral operator f �→ ∑

x∈E f(x)k(x, y). Itis this interpretation of the Martin kernel that has a natural noncommutativeanalogue.

9.4 The Martin Kernel for the Quantum Random

Walk

We consider a central quantum random walk, therefore we have a positivedefinite central function φ on SU(2) such that φ(e) ≤ 1, and the associated

Introduction to Random Walks on Noncommutative Spaces 93

state ν on A(SU(2)). We have seen that it has an associated transition kernelgiven by the completely positive map Φν = (I ⊗ ν) ◦ ∆. This map acts asλg �→ φ(g)λg on the generators λg of the von Neumann algebra A(SU(2)). Weknow that if φ(e) = 1 then there exists only one φ-harmonic weight, namelythe one given by the function 1. In this case we expect that the Martinboundary will be given by a one point compactification, which consists justin adding a unit to the algebra, so in order to avoid this case, we shall assumehere that φ(e) < 1.

We can consider the associated potential which is equal to U =∑∞

n=0 Φnν ,

and which acts by multiplication by the function∑∞

n=0 φn = 11−φ . We shall

consider the action of the potential on the space of coefficients i.e. the di-rect sum ⊕χMχ where each element of this space can be identified with apolynomial function on SU(2) acting by convolution, i.e. by the operator∫

SU(2) p(g)λgdg. Then the Martin kernel will be defined, by analogy with the

case of classical Markov chains, by

p �→∫

SU(2)p(g)

1−φ(g)λgdg∫

SU(2)1

1−φ(g)λgdg

Note that the operator∫

SU(2)1

1−φ(g)λgdg lies in the center of A(SU(2)),

therefore there is no ambiguity in the quotient of the preceding formula.

9.5 Pseudodifferential Operators of Order Zero

and the Martin Compactification

In order to find the Martin compactification of the quantum random walk,we shall identify the C∗-algebra generated by C∗(SU(2)) and by the image ofthe Martin kernel, and show that it gives rise to a three terms exact sequence.We will first exhibit a certain exact sequence

0 → C∗(SU(2)) → M → C(S2) → 0 (9.1)

where M is our sought for Martin compactification, and C(S2) is the algebraof continuous functions on the two dimensional sphere S2. For this we considerthe three operators in A(SU(2)) associated with Pauli matrices, which we callX, Y, Z, and the Casimir operator C = X2+Y 2+Z2+I which acts by (n+1)2

on the space of coefficients of the n-dimensional representation of SU(2), andbuild three operators

x = XC−1/2, y = Y C−1/2, z = ZC−1/2

Clearly these operators are self-adjoint and bounded. Using the commutationrelations (3.1) one can see that [x, y], [x, z], [y, z] and x2 + y2 + z2 − I are

94 P. Biane

compact operators on L2(SU(2)). It follows that there exists a map fromthe algebra generated by x, y, z to the algebra of polynomial function onthe sphere, sending x, y, z to the three coordinate functions, and this mapvanishes on the compact operators. Actually this map extends by continuityto the C∗ algebra generated by x, y and z and yields the exact sequence (9.1).One can interpret also the algebra M as the algebra of right invariant pseudodifferential operators of order zero on SU(2), then the map M → C(S2) of(9.1) is the principal symbol map (see [B5]). Once we have introduced theexact sequence above, we can state the theorem which is the analogue, forcentral states on the dual of SU(2), of the Ney-Spitzer theorem.

Theorem 9.1. The C∗ algebra generated by the image of the Martin kernelis the algebra M. The Martin kernel yields a section K : C(S2) → M.

The proof of the theorem relies on a detailed analysis of the Clebsch-Gordanformulas, see [B5].

Recently the problem of the Martin or Poisson boundary have been con-sidered for quantum groups, see [Col], [I], [INT]

10 Central Limit Theorems for the Bernoulli RandomWalk

Just as in the classical case there exists central limit theorems for theBernoulli random walk, however the noncommutativity here plays an im-portant role, and according to whether the state we chose is central or notthe limit is quite different.

10.1 The Case of a Central State

We consider the triple of processes (Xn, Yn, Zn)n≥1 constructed in section(3.1). We use the tracial state to contruct the product, M2(C)⊗∞ which isthus endowed with the tracial state σ = (1

2Tr)⊗∞. We renormalize the threeprocesses according to

X(λ)t =

X[λt]√λ

, Y(λ)t =

Y[λt]√λ

, Z(λ)t =

Z[λt]√λ

where [x] is the integer part of x. This triple of processes converges whenλ → ∞, in the sense of moments, towards a three dimensional Bownianmotion.

Theorem 10.1. Let (Xt, Yt, Zt)t≥0 be a three dimensional Brownian motion,then for any polynomial in 3n noncommuting indeterminates P , and all timest1, . . . , tn, one has

Introduction to Random Walks on Noncommutative Spaces 95

limλ→∞ σ(P (X(λ)t1 , Y

(λ)t1 , Z

(λ)t1 , . . . , X

(λ)tn

, Y(λ)tn

, Z(λ)tn

)) =

E[P (Xt1 , Yt1 , Zt1 , . . . , Xtn , Ytn , Ztn)]

Let us sketch the proof of this result. It is enough to prove the result formonomials. First we see that for any real coefficients x, y, z the process

xX(λ)t + yY

(λ)t + zZ

(λ)t converges towards real Brownian motion. By po-

larization this implies that for any monomial in X, Y, Z the sum over allmonomials with the same total partial degrees in X, Y, Z converges towards

the required limit. Consider two monomials of the form M1X(λ)t Y

(λ)s M2 and

M1Y(λ)s X

(λ)t M2, their difference is, thanks to the commutation relations,

M1Z(λ)s∧tM2/

√λ. This is a monomial of smaller degree, with a factor 1/

√λ. We

conclude, by induction on degrees of monomials, that the difference betwen

the expectations of the two monomials M1X(λ)t Y

(λ)s M2 and M1Y

(λ)s X

(λ)t M2

converges to 0 as λ → ∞. It follows that the expectations of all monomialswith the same partial degrees in variables converge to the same limit. ⊓⊔

We observe that the spin process, normalized by S[λt]/√

λ converges indistribution to a three dimensional Bessel process as λ → ∞.

10.2 The Case of a Pure State

Now we consider the quantum Bernoulli random walk with the pure stategiven by the vector e0. We shall consider the convergence of the moments ofthe triple of processes

(X(λ)t , Y

(λ)t , Z

(λ)t ) = (

X[λt]√λ

,Y[λt]√

λ,Z[λt]

λ); t ≥ 0.

We shall prove that there exists operators (Xt, Yt, Zt)t≥0 on some Hilbertspace H with a vector Ω ∈ H , such that for every polynomial in noncom-muting indeterminates P (Xt1 , Yt1 , Zt1 , . . . , Xtn , Ytn , Ztn) one has

limn→∞〈P (X(λ)t1 , Y

(λ)t1 , Z

(λ)t1 , . . . , X

(λ)tn

, Y(λ)tn

, Z(λ)tn

)e∞0 , e∞0 〉) =

〈P (Xt1 , Yt1 , Zt1 , . . . , Xtn , Ytn , Ztn)Ω, Ω〉

For this we shall first investigate the case where n = 1, t1 = 1, thus we havejust three operators (Xn√

n, Yn√

n, Zn

n ) and let n → ∞. Let H be a Hilbert space

with a countable orthonormal basis ε0, . . . , εn, . . ., and let us define operators,with domain the algebraic sum ⊕n

i=0Cεi, by the formula

a+(εi) =√

j + 1 εj+1

a−(εj) =√

j εi−1(10.1)

96 P. Biane

Theorem 10.2. For any polynomial in three noncommutative indetermi-nates one has

limn→∞

〈P (Xn√

n,

Yn√n

,Zn

n)e⊗n

0 , e⊗n0 〉 = 〈P (a+ + a−,

1

i(a+ − a−), I)ε0, ε0〉

The proof is immediate by inspection of the formula (3.2) and comparisonwith (10.1). ⊓⊔

Observe that one has the adjointness relations

〈a+u, v〉 = 〈u, a−v〉

for all u, v in the domain. It follows that the operators a++a− and 1i (a

+−a−)are unbounded symmetric operators on H , and thus are closable. We will seebelow that they have self-adjoint extensions. They satisfy the commutationrelation

[a+, a−] = −I

on their common domain, spanned by the vectors εi.The operators a+, a− thus obtained are well known under the name of

creation and annihilation operators for the quantum harmonic oscillator. Onecan give a natural model for these operators using a gaussian random variable.For this, remark that the distribution of the operator a+ + a− is gaussian.This follows easily from the fact that each Xn follows a standard binomialdistribution, and the convergence of this binomial distribution to the gaussiandistribution, by the de Moivre-Laplace theorem. This property of the operatora+ + a−, and the fact that ε0 is a cyclic vector for a+ + a−, i.e. the vectors(a+ + a−)nε0 span a dense subspace of H , allows us to identify the spaceH in a natural way with the L2 space of a gaussian random variable. Thevectors εn can be obtained from the vectors (a++a−)n by the Gram-Schmidtorthogonalization procedure. It is well known that, for a gaussian variable X ,the polynomials obtained by the Gram-Schmidt orthonormalization processfrom the sequence Xn are the Hermite polynomials. Thus when we identifythe space H with the L2 space of the gaussian measure on R the vectors εn

become identified with the Hermite polynomials. Then the operator a− isidentified with d

dx and the operator a+ with x − ddx .

The product a+a− has eigenvalues 0, 1, 2, . . . corresponding to the the re-spective eigenvectors ε0, ε1, . . .. It is known as the number operator in quan-tum field theory.

We will in the next section systematize the construction above and showhow to deduce the limit of the renormalized quantum random walk to aquantum Brownian motion.

Introduction to Random Walks on Noncommutative Spaces 97

10.3 Fock Spaces

Let H be a complex Hilbert space. For each integer n the symmetric groupSn acts on H⊗n by permutation of the factors in the tensor product.

Definition 10.3. We denote H◦n the subspace of H⊗n formed by vectorsinvariant under the action of Sn.

If h1, . . . , hn ∈ H then we let

h1 ◦ . . . ◦ hn =1√n!

∑

σ∈Sn

hσ(1) ⊗ . . . ⊗ hσ(n)

which is a multiple of the orthogonal projection of h1⊗ . . .⊗hn on H◦n. Onehas

〈h1 . . . ◦ hn, h′1 ◦ . . . ◦ h′

n〉 =∑

σ∈Sn

n∏

i=1

〈hi, h′σ(i)〉

The Fock space built on H is the Hilbert space direct sum

Γ (H) = ⊕∞n=0H

◦n

where H◦0 is a one dimensional Hilbert space spanned by a unit vectorΩ, called the vacuum vector of the Fock space. The algebraic direct sum⊕∞

alg,n=0H◦n, denoted by Γ0(H), is a dense subspace of Γ (H).

For every h ∈ H we define the exponential vector associated with h by

ξ(h) =

∞∑

n=0

h◦n

n!

one has〈ξ(h), ξ(h′)〉 = e〈h,k〉

furthermore the vectors ξ(h); h ∈ H form a linearly free subset, generatingalgebraically a dense subspace of Γ (H).

If the space H is written as the orthogonal direct sum of two Hilbertsubspaces H = H1 ⊕ H2, then there is a canonical isomorphism

Γ (H) ∼ Γ (H1) ⊗ Γ (H2) (10.2)

which can be obtained, for example, by identifying the exponential vectorξ(v1+v2) ∈ Γ (H), where v1 ∈ H1 and v2 ∈ H2, with the vector ξ(v1)⊗ξ(v2) ∈Γ (H1) ⊗ Γ (H2).

Let h ∈ H , we define two operators on the domain Γ0(H) by

a+h (h1 ◦ . . . ◦ hn) = h ◦ h1 ◦ . . . ◦ hn

a−h (h1 ◦ . . . ◦ hn) =

∑ni=1〈hi, h〉h1 ◦ . . . ◦ hi ◦ . . . ◦ hn

98 P. Biane

Operators of the form a+h are called creation operators while the a−

h arecalled annihilation operators. When H is one dimensional, generated by aunit vector u there is a natural identification of Γ (H) with the Hilbert spaceof the preceding section where the vector u◦n is identified with

√n!ǫn. Then

the operators a+u and a−

u coincide with a+ and a−.Another operator of interest on Γ (H) is the number operator Λ which has

eigenvalue n on the subspace H◦n. If (ei)i∈I is an orthonormal basis in H ,then the number operator Λ has the expansion

Λ =∑

i

a+e1

a−ei

.

One can see that the creation and annihilation operators satisfy the adjointrelation

〈a+h u, v〉 = 〈u, a−

h v〉 h, u, v ∈ Γ0(H)

as well as the commutation relation

[a+h , a−

k ] = −〈h, k〉I

on the domain Γ0(H). In particular they are closable, and it is easy to seethat the exponential vectors belong to the domain of their closure, with

a+h ξ(h′) =

d

dtξ(h′ + th)t=0 a−

h ξ(h′) = 〈h′, h〉ξ(h) (10.3)

We shall see that the real part of the creation operator Ph = a+h +a−

h has a self-adjoint extension, as well as its imaginary part Qh. For this we consider thefollowing Weyl operators, given on the vector space generated by exponentialvectors by the formula

Wuξ(h) = ξ(h + u)e−〈h,u〉− 12 〈u,u〉

It is easy to check that

〈Wuξ(h), Wuξ(h′)〉 = 〈ξ(h), ξ(h′)〉 (10.4)

for all u, v, h, h′ ∈ H , therefore the operators Wu extend to unitary operatorson Γ (H), furthermore

WuWv = Wu+ve−iℑ〈u,v〉 (10.5)

and for any u ∈ H the operators (Witu; t ∈ R) form a one parameter groupof unitary operators, whose generator is given by Pu on exponential vectors.Similarly, the operators (Wtu; t ∈ R) form a one parameter group of unitaryoperators, whose generator is given by Qu, and more generally for θ ∈ [0, 2π[the vectors (Weiθ tu; t ∈ R) form a one parameter group of unitary operators,whose generator is given by cos θQu + sin θPu.

Introduction to Random Walks on Noncommutative Spaces 99

It follows from Stone’s theorem that Pu, Qu, and all their linear combina-tions, have a self adjoint extension. These operators satisfy the commutationrelation

[Pu, Qv] = −2iℜ〈u, v〉.We observe that if H splits as an orthogonal direct sum H = H1 ⊕ H2 andu ∈ H1, then the operator Wu admits a decomposition Wu = Wu ⊗ I in thedecomposition (10.2).

Let now K ⊂ H be a real Hilbert subspace such that ℑ〈u, v〉 = 0 forall u, v ∈ K (if K is maximal, it is called a Lagrangian subspace), then by(10.5) the unitary operators Wiu; u ∈ K form a commutative family, and thegenerators Pu, u ∈ K of the one parameter subgroups (Witu; t ∈ R) forma commuting family of self-adjoint operators with common dense domainΓ0(H). We can therefore investigate the joint distribution of these operators.

Proposition 10.4. The operators Pu, for u ∈ K form a gaussian family withcovariance 〈Pu, Pv〉 = 〈u, v〉.

The operators Qu, for u ∈ K form a gaussian family with covariance〈Qu, Qv〉 = 〈u, v〉.

For the proof it is enough to prove that any linear combination of theseoperators has a gaussian distribution with the right variance, i.e. that Pu isa gaussian with variance 〈u, u〉. For this one computes the Fourier transform

〈eiPuΩ, Ω〉 = 〈eiPuξ(0), ξ(0)〉 = 〈ξ(u)e−12 〈u,u〉, ξ(0)〉 = e−

12 〈u,u〉

The proof for the operators Qu is similar. ⊓⊔We let now H = L2(R+), and take as Lagrangian subspace the subspace of

real valued functions. Then the family (Pt := P1[0,t]; t ≥ 0) has the covariance

〈Pt, Ps〉 = s∧ t, and thus has the distribution of a real brownian motion. Thesame is true of the operators Qt := 1

i (a+1[0,t]

− a−1[0,t]

) which form another

Brownian motion satisfying the commutation relations

[Pt, Qt] = 2it

We shall call the pair (Pt, Qt)t≥0 of continuous time processes a noncommu-tative brownian motion.

We can now state the limit result we had in view in the beginning of thissection.

Theorem 10.5. For any polynomial P in noncommuting variables, one has

limλ→∞〈(P (X(λ)t1 , Y

(λ)t1 , Z

(λ)t1 , . . . , X

(λ)tn

, Y(λ)tn

, Z(λ)tn

)e∞0 , e∞0 〉) =〈P (Pt1 , Qt1 , t1.I, . . . , Ptn , Qtn , tn.I))Ω, Ω〉]

The proof is an elaboration of the proof we gave for one time.

100 P. Biane

We shall see in the next section that, again, one can interpret this pair ofprocesses as a noncommutative Markov process with values in a noncommu-tative space.

The noncommutative Brownian motion is the basis of a theory of noncom-mutative stochastic integration which has been developped by Hudson andParthasarathy, see e.g. [Pa].

11 The Heisenberg Group and the NoncommutativeBrownian Motion

The Heisenberg group is the set H = C × R endowed with the group law

(z, w) ⋆ (z′, w′) = (z + z′, w + w′ + ℑ(zz′))

This is a nilpotent group, its center being {0}×R, and the Lebesgue measureon C × R is a left and right Haar measure for this group.

The Weyl operators defined in the preceding section on a Fock space Γ (C)define unitary representations of the group H, by setting, for τ ∈ R∗,

ρτ (z, w) = Wzτ1/2eiτw

if τ > 0 andρτ (z, w) = Wz|τ |1/2eiτw

if τ < 0.Another family of representations is given by the one dimensional charac-

tersρξ(z, w) = eiℜ(zξ)

for ξ ∈ C.All these representations are irreducible, are non equivalent and they ex-

haust the family of equivalence classes of irreducible representations of H.We consider the C∗ algebra of H, which is the C∗-algebra generated by

the convolution algebra L1(H) on L2(H). This algebra is a sub C∗-algebra ofB(L2(H)). Let us denote z = q + ip then the Lie algebra of H is composedof the vector fields

∂

∂w;

∂

∂q+ p

∂

∂w;

∂

∂p− q

∂

∂w.

We shall denote by iT, iQ, iP the unbounded operators on L2(H), affiliated toC∗(H), which correspond to these vector fields. Thus P, Q, T are unboundedself-adjoint operators, which satisfy the commutation relation

[P, Q] = −2iT.

Introduction to Random Walks on Noncommutative Spaces 101

The plane T=0

T

T=t

[P,Q]=it

The "quantum plane"

Fig. 5 The dual of the Heisenberg group.

As in the case of SU(2) one can give a heuristic description of the non-commutative space dual to H using the generators of the Lie algebra of H,which define three noncommuting unbounded self-adjoint operators P, Q, T .We think of these operators as coordinate functions on this dual space, sat-isfying the commutation relations

[P, Q] = −2iT [P, T ] = [Q, T ] = 0.

Since the coordinate T belongs to the center, it allows to decompose thespace into slices according to the values of this coordinate. When T = 0,the coordinates P and Q commute, and the corresponding slice is a usualplane, with two real coordinates. This corresponds to the one dimensionalrepresentations of the group. When T = τ a non zero real number, thetwo coordinates P, Q generate a von Neumann algebra isomorphic to B(H),and corresponding to the irreducible representation sending T to τI. Notethat in this representation the operator P 2 + Q2 has a discrete spectrum2|τ |, 6|τ |, 10|τ |, . . ..

Let us consider, for t ≥ 0, the functions on H

ϕ±t (z, w) = 〈ρ±t(z, w)Ω, Ω〉 = e±itw− 1

2 tzz .

By construction, these functions are positive definite functions on H, andform two multiplicative semigroups. To these functions correspond convo-lution semigroups of states, and semigroups of completely positive maps.

102 P. Biane

The semigroup of noncommutative brownian motion on the dual of H isthe associated semigroup of completely positive maps on C∗(H). Recallthat this semigroup is obtained by composing the coproduct ∆ : C∗(H) →M(C∗(H)) ⊗ M(C∗(H)) (recall that M(A) is the multiplier algebra of A)with the state associated with the function. Thus

Φ±t = (νϕ±

t⊗ I) ◦ ∆

or equivalentlyΦ±

t (λg) = ϕ±t (g)λg for g ∈ H.

Let ν be a state on C∗(H), and ρν : C∗(H) → B(Hν) be the associatedGNS representation. We consider the two families of homomorphisms

j±t : C∗(H) → B(Hν ⊗ Γ (L2(R+)))j±t (z, w) = ρν(z, w) ⊗ Wz1[0,t]

e±itw

We shall prove that these homomorphisms constitute dilations of some com-pletely positive convolution semigroups on C∗(H). For each time t ≥ 0 wehave a direct sum decomposition L2(R+) = L2([0, t]) ⊕ L2([t, +∞[) and acorresponding factorization

Γ (L2(R+)) = Γ (L2([0, t])) ⊗ Γ (L2([t, +∞[))

Accordingly for each t > 0 there are subalgebras

Wt = B(Γ (L2([0, t]))) ⊗ I ⊂ B(Γ (L2(R+)))

and linear maps

Et := Id ⊗ 〈.Ω[t, Ω[t〉 : B(Γ (L2(R+))) → Wt

where Ω[t is the vacuum vector of the space Γ (L2([0, t])).

Lemma 11.1. For each t ≥ 0 the map Et is a conditional expectation withrespect to the state 〈.Ω, Ω〉.

Indeed if a ∈ B(L2(R+)) has a decomposition a = at] ⊗ a[t then one has

Et(a) = at] ⊗ 〈a[tΩ[t, Ω[t〉

and for b, c ∈ B(L2([0, t]))

〈bacE(u), E(v)〉 = 〈bEt(a)cE(u1[0,t]), E(v11[0,t])〉〈a[tE(u1[t,+∞[), E(v1[t,+∞[)〉= 〈bEt(a)cE(u), E(v)〉

One checks easily that the homomorphism j±t sends C∗(H) to Wt, further-more we have

Introduction to Random Walks on Noncommutative Spaces 103

Proposition 11.2. The maps (j±t , Et, Wt) form a dilation of the completelypositive semigroup Φ±

t , with initial distribution ν.

Proof. This is a bookkeeping exercise using the definition of the Weyl oper-ators, one has to check that, for t, s ≥ 0 and u, v ∈ L2(R+), one has

〈j±t+s(z, w)E(u1[0,t]), E(v1[0,t])〉 = e±isw− 12 s|z|2〈j±t (z, w)E(u1[0,t]), E(v1[0,t])〉

⊓⊔For every one parameter subgroup of H, there is a completely posi-

tive semigroup given by restriction of (Φ±t )t≥0. For subgroups of the form

(xz, 0); u ∈ R, with z ∈ C∗ isomorphic to R, one sees that the semigroup isthat of Brownian motion on the dual group. Thus we recover the Brownianmotions (Pt, Qt) of the preceding section by looking at jt(P ) and jt(Q).

The restriction to the center (0, w); w ∈ R gives a semigroup of uniformtranslation on the real line.

11.1 The Quantum Bessel Process

Bessel processes are radial parts of Brownian motions. Here we shall exploitthe action of the unitary group U(1) on the Heisenberg group in order to findan abelian algebra which is left invariant by the semigroup and study theMarkov process associated with the restriction of Φ±

t to this subalgebra. Leteiθ be a complex number with modulus 1, then there exists an automorphismaθ of the Heisenberg group defined by

aθ(z, w) = (eiθz, w)

and this automorphism extends to an automorphism of the C∗ algebra.

Proposition 11.3. The subalgebra of C∗(H) composed of elements invariantunder the above action of U(1) is an abelian C∗ algebra.

The characters of this abelian C∗-algebra have been computed by A. Koranyi,they are given by the formula

χ(f) =

∫

Hω(g)f(g)dg

for f ∈ L1(H), invariant under the action of U(1), where ω belongs to theset of functions

{ωτ,m|τ ∈ R∗, m ∈ N} ∪ {ωμ|μ ∈ R+}

104 P. Biane

Fig. 6 The Heisenberg fan.

withωτ,m(z, w) = meiτw− 1

2 |τ ||z|2

Lm(|τ ||z|2)ωμ(z, w) = j0(μ|z|2)

Here Lm are the Laguerre polynomials defined by the generating series

∞∑

m=0

Lm(x)tm =e−

xt1−t

1 − t

and j0 is the usual Bessel function.The spectrum of the algebra C∗

R(H) can be identified with a closed subsetof R2, which consists in the union of the halflines {(x, kx); x > 0} for k ∈ N,the halflines (x, kx); x < 0 for k ∈ N

∗, and the halfline (0, y); y ≥ 0.It is the spectrum of the unbounded operator 1

2 (P 2 + Q2 − T ), and thisalgebra is the algebra of functions of this operator.

The picture gives a “fan” consisting of a union of halflines originating from0 as depicted below.

We shall call noncommutatif Bessel semigroup the restriction of Φ±t to the

abelian subalgebra C∗R(H). In order to compute this semigroup we need,

for each character ω to decompose the functions ωϕt into an integral ofcharacters. The result is given by the following.

Proposition 11.4. The noncommutative Bessel semigroup Φ+t is given by

the following kernel.If x = (σ,−kσ) with σ < 0 and τ = σ + t then

qt(x, dy) =∞∑

l=k

(l − 1)!

(k − 1)!(l − k)!(1 − τ

σ)l−k(

τ

σ)kδ(τ,−lτ)(dy)

Introduction to Random Walks on Noncommutative Spaces 105

Fig. 7

If x = (σ,−kσ) with σ < 0 and 0 = σ + t, (y = 0, r) then

qt(x, dy) =( r

t )k−1

(k − 1)!e−

rt δ0 ⊗ dr

If x = (σ,−kσ) with σ < 0 and τ = σ+ > 0 then

qt(x, dy) =

∞∑

l=0

(l + k − 1)!

(k − 1)!l!(1 − τ

t)l+k(

τ

t)kδ(τ,lτ)(dy)

If x = (0, r) then

qt(x, dy) =( r

t )k−1

l!e−

tlt δt,lt(dy)

If x = (σ, kσ) with σ > 0 and τ = σ + t

qt(x, dy) =

l=k∑

l=0

(l − 1)!

(k − 1)!(l − k)!(1 − σ

τ)k−l(

σ

τ)lδ(τ,lτ)(dy)

The computations can be found in [B6].A typical trajectory of the process is depicted in the above picture. It

starts from a point (σ,−σ) with σ < 0. During the whole process the firstcoordinates follows a uniform translation to the right. The trajectory startson the line y = −x, and with an intensity dt

−σ+t , then jumps to the line(y = −2x), which it follows before jumping to the next line (y = −3x) withan intensity 2dt

−σ+t , and so on, until it reaches after infinitely many jumps theline x = 0, then the process on the right half plane does the jumps from theline (y = kx) to (y = (k− 1)x), until it finally reaches the line y = 0 where itstays forever. One can actually construct this process from a birth and deathprocess known as the Yule process, and the embedding of this process intothe Heisenberg fan yields a construction of the space-time boundary of thisprocess, see [B6].

106 P. Biane

12 Dilations for Noncompact Groups

12.1 The General Case

We shall now extend the preceding construction to the case of arbitrary lo-cally compact groups. Let G be a locally compact group, with right Haarmeasure m, and consider its convolution algebra L1(G). We endow this alge-bra with the norm ‖f‖ = sup ‖ρ(f)‖ where the supremum is over all unitaryrepresentations ρ of the group G, extended to the algebra L1(G). This yieldsthe full C∗-algebra of G, denoted C∗(G). When the group G is amenable, forexample if G is compact or for the Heisenberg group which we have met be-fore, this coincides with the completion of the action of the group on L2(G),which is called the reduced C∗-algebra of G. When the group is nonamenable,C∗

r (G) is strictly smaller than C∗(G). A continuous positive definite functionon G, such that ϕ(e) = 1 defines a state as well as a completely positivecontraction on C∗(G), whose restriction to L1(G) is given by f �→ fϕ. Letnow ψ be a continuous, conditionally negative definite function on G, withψ(e) = 0. Recall that this means that for all t ≥ 0 the function e−tψ is apositive definite function on G, or equivalently, by Schonberg’s theorem, thatfor all z1, . . . , zn ∈ C with

∑

i zi = 0, and g1, . . . , gn ∈ G one has

∑

ij

zizjψ(g−1j gi) ≤ 0.

There is an associated semigroup of completely positive contractions onC∗(G). We shall now, following Parthasarathy and Schmidt [PS], constructa dilation of this semigroup. Let ν be a state on C∗(G) which will be theinitial state. The GNS construction yields a unitary representation π of G ona Hilbert space Hπ, and η ∈ Hπ such that ν(x) = 〈π(x)η, η〉 for x ∈ C∗(G).A variant of the GNS construction associates to the function ψ a unitaryrepresentation of G in a Hilbert space Hψ, and a cocycle v : G → Hψ for thisrepresentation. Thus v is a continuous function which satisfies

v(gh) = gv(h) + v(g)

and〈v(g), v(h)〉 = −ψ(h−1g) + ψ(g) + ψ(h−1) − ψ(e) (12.1)

for all g, h ∈ G. Conversely, any function ψ satisfying the above equation forsome representation and cocycle v is conditionally negative definite. Indeedone has

∑

ij

zizjψ(g−1j gi) = −‖

∑

j

zjv(gj)‖2 ≤ 0

if∑

j zj = 0.

Introduction to Random Walks on Noncommutative Spaces 107

Let Γ = Γ (L2(R+) ⊗ Hψ), and let W = B(Hπ ⊗ Γ ) ∼ B(Hπ) ⊗ B(Γ ).Let ω be the pure state on W associated with the vector η ⊗ Ω. One candefine the subalgebras Wt = B(Hπ ⊗Γt)⊗ Id associated with the orthogonaldecomposition L2(R+) ⊗ Hψ = (L2([0, t]) ⊗ Hψ) ⊕ (L2([t, +∞[) ⊗ Hψ), andthe conditional expectations Et : W → Wt with respect to the state ω. Onedefines a unitary representation of G on exponential vectors by

V t(g)(E(u)) = etψ(g)+〈1[0,t]⊗vt(g),u〉E(U t(g)(u) + 1[0,t] ⊗ vt(g))

One thus gets a representation of G on Hπ ⊗Γ by taking the tensor productof V t with the representation π, and this yields a family of morphisms jt :C∗(G) → W .

Proposition 12.1. The family (jt,W ,Wt, Et, ω) forms a dilation of the com-pletely positive semigroup, with initial distribution ν.

The proof is a bookkeeping exercise. This construction has been extendedby Schurmann to a class of bialgebras, see [Sc], allowing him in particular togive a nice construction of the Azema martingales.

12.2 Free Groups

Let now Fn be a free group on n generators g1, . . . , gn. Each element of Fn

can be written in a unique way as a reduced word w = gε1

i1. . . gεk

ik, where

one has εj = ±1 for all j and i1 �= i2 �= i3 . . . ik−1 �= ik. For such an elementone defines its length l(w) = k. This is the smallest integer k such that w canbe expressed as a product of k elements in the set {g1, g

−11 , g2, . . . , gn, g−1

n }.

Proposition 12.2. (Haagerup [H]) The function l is conditionally negativedefinite on Fn.

Proof. Consider the Cayley graph of Fn built on the generators. Thus thisgraph has as vertices the elements of Fn and its edges are the pairs {g, h}such that h−1g is a generator or the inverse of a generator. This Cayley graphis a regular tree in which each vertex has 2n neighbours. For any g ∈ Fn onecan consider the unique shortest path in the graph between Id and g. Let En

be the set of edges of the Cayley graph, endowed with the counting measure,then one defines v(g) ∈ L2(En) to be the indicator function of this shortestpath from Id to g in the Cayley graph. Thus v(g)(e) = 1 if and only if theedge e is on the shortest path from Id to g. One can easily check, using theproperties of trees that for any h, h ∈ Fn one has

l(g) + l(h) − l(h−1g) = 2〈v(g), v(h)〉L2(Fn)

108 P. Biane

indeed this scalar product counts the number of common edges in the shortestpaths from Id to g and h. This implies, by (12.1) that l is conditionallynegative definite. ⊓⊔

It follows from the previous proposition that there exists, on the full C∗

algebra of Fn, a semigroup of unit preserving completely positive maps, givenby the formula

Φt(λg) = e−tl(g)λg

The theory of the previous section allows us to construct a dilation of thissemigroup. As before we shall be interested in the restriction of this com-pletely positive semigroup to commutative subalgebras. The first one will bethe subalgebra of the subgroup generated by one of the generators. Let gi

be this generator, then this subgroup is isomorphic to Z by k �→ gki , therefore

its dual is isomorphic to the group of complex numbers of modulus 1. Therestriction gives us a Markov semigroup on the group of complex numbers ofmodulus 1. This semigroup is easy to characterize, it sends the function eikθ

on the unit circle to the function e−|k|+ikθ. In other words this is the integraloperator on the unit circle given by the Poisson kernel

Pt(θ, θ′) =

1 − e−2t

1 − 2e−t cos(θ − θ′) + e−2t.

This is a convolution semigroup, as expected.The other commutative algebra of interest is the algebra R(Fn) consist-

ing of radial elements. It is generated by the elements χl =∑

l(g)=l λg forl = 0, 1, . . . , and it is immediate that the completely positive semigroup as-sociated with the length function leaves this algebra invariant. Actually onehas Φt(χl) = e−tlχl. These elements satisfy the relations

χ0 = I

χ21 = χ2 + 2nχ0

χlχl = χl+1 + (2n − 1)χl−1 l ≥ 2

From this we conclude that R(Fn) is the commutative von Neumann algebragenerated by the self-adjoint element χ1, and its spectrum is the spectrum ofχ1. In order to compute the norm of χl we need just to consider the trivialrepresentation of Fn in which all gi are mapped to the identity, and we get‖χl‖ = 2n(2n − 1)l−1 for l ≥ 1, the number of elements of length l in Fn.Any character ϕ : R(Fn) → C is determined by its values on χ1. For such acharacter ϕ, with ϕ(χ1) = x, one has

ϕ(χ0) = 1; ϕ(χ2) = x2−2n; ϕ(χl+1) = xϕ(χl)− (2n−1)ϕ(χl−1) for l ≥ 2

from which one infers that

Introduction to Random Walks on Noncommutative Spaces 109

ϕ(χl) =λl+1

1 − λl+12

λ1 − λ2− λl−1

1 − λl−12

λ1 − λ2for l ≥ 1 (12.2)

where λ1, λ2 are the two roots of the equation λ2 − xλ + 2n − 1 = 0.We verify that the character ϕx defined by the formula above is real and

bounded if and only if x ∈ [−2n, 2n]. The spectrum of the algebra R(Fn)thus coincides with the interval [−2n, 2n], and the element χl corresponds toa polynomial function Pl(χ1) = χl, where the polynomials are determined bythe recursion

P0 = 1, P1(x) = x, xPl(x) = Pl+1(x) + (2n − 1)Pl−1.

This three term recursion relation is characteristic of a sequence of orthogonalpolynomials. The orthogonalizing measure is the distribution of χ1 in thenoncommutative probability space (A(Fn), δe) where δe is the pure state,in the left regular representation of Fn, corresponding to the identity. Thusδe(λg) = 1 if g = e and δe(λg) = 0 if not. This measure is known as theKesten measure (see [Ke]) and has the density

dmn(x) =2n

2n − 14π

√

4(2n − 1) − x2

4n2 − x2

on the interval [−2√

2n− 1, 2√

2n − 1]. The discrepancy between the interval[−2n, 2n] which is the spectrum of χ1 and the support of the measure mn

comes from the fact that Fn is a nonamenable group and therefore some ofits unitary representations are not weakly contained in the regular represen-tation.

The semigroup of the restriction of (Φt)t≥0 to the subalgebra R(Fn)sends the polynomial function Pl(x) on the interval [−2n, 2n] to the functione−tlPl(x). We can compute the transition probabilities pt(x, dy) by findingthe integral representation

e−tlPl(x) =

∫

[−2n,2n]

pt(x, dy)Pl(y)

for each x ∈ [−2n, 2n].If x belongs to the support of the Kesten measure, then since the poly-

nomials Pl; l ≥ 0 form an orthogonal basis of the L2 space of the Kestenmeasure, and ‖Pl‖2

2 = 2n(2n − 1)l one obtains pt through the orthogonalexpansion

pt(x, dy) = e−tdmn(y) +∞∑

l=1

e−tl 1

2n(2n − 1)l−1Pl(x)Pl(y)dmn(y)

When x is outside this support, then by (12.2) the sequence Pl(x) is un-

bounded and has exponential growth of rate ξ =x+

√x2−4(2n−1)

2 and the

110 P. Biane

character value e−tlPl(x) has the asymptotic behaviour e−tlξl as l → ∞,and if e−tξ > 2

√2n − 1, then let x(t) = e−tξ + (2n − 1)et/ξ. The func-

tion e−tlPl(x) can be written as is a linear combination of Pl(x(t)), whichpicks up the dominant term as l → ∞ and the Pl(y) for y in the interval[−2

√2n − 1, 2

√2n − 1]. More precisely the quantity

Qtl(x) = e−tlPl(x) − e−tξ − (2n − 1)et/ξ

e−tξ − et/ξ

ξ − (2n − 1)/ξ

ξ − 1/ξPl(x(t))

decreases exponentially as l → ∞, and thus one has

pt(x, dy) = ctδx(t) +∞∑

l=1

e−tl 1

2n(2n− 1)l−1(Pl(x) − ctPl(x(t)))Pl(y)dmn(y)

with ct = e−tξ−(2n−1)et/ξe−tξ−et/ξ

ξ−(2n−1)/ξξ−1/ξ . If ξ(t) < 2

√2n − 1 then there is a simi-

lar decomposition, but the term ct is 0.Thus the process starting form a point x ∈ [−2n, 2n] \ [−2

√2n − 1,

2√

2n− 1] performs a translation towards the central interval, and at somepoint jumps into it, and after that performs a certain pure jump processinside the interval [−2

√2n − 1, 2

√2n − 1] where it remains forever.

13 Pitman’s Theorem and the Quantum Group SUq(2)

13.1 Pitman’s Theorem

Let (Bt)t≥0 be a real Brownian motion, with B0 = 0, and let

St = sup0≤s≤t

Bs

then Pitman’s theorem states that the stochastic process

Rt = 2St − Bt; t ≥ 0

is a three dimensional Bessel process, i.e. is distributed as the norm of athree dimensional Brownian motion. There is a discrete version of Pitman’stheorem, actually it is this discrete version that Pitman proved in his originalpaper [Pi]. We start from a symmetric Bernoulli random walk Xn = x1+. . .+xn where the xi are i.i.d. with P (xi) = ±1 = 1/2, and build the processesSn = max1≤k≤n Xk, and Tn = 2Sn−Xn. Pitman proved in [Pi] that (Tn)n≥0

is a Markov chain on the nonegative integers, with probability transitions

Introduction to Random Walks on Noncommutative Spaces 111

Fig. 8

p(k, k + 1) =k + 2

2(k + 1)p(k, k − 1) =

k

2(k + 1)

and the theorem about Brownian motion can be obtained by taking theusual approximation of Brownian motion by random walks. We see that thisMarkov chain is, up to a shift of 1 in the variable, exactly the one that weobtained in Theorem 3.2 when considering the spin process. This is not acoincidence as we shall see, actually we will understand this connection byintroducing quantum groups in the picture. Before that, let us give the proofof Pitman’s theorem We consider the stochastic process ((Sn, Xn);n ≥ 1),with values in {(s, k) ∈ N × Z | s ≥ k}. It is easy to see that this stochasticprocess is a Markov chain, with transition probabilities

p((s, k), (s, k + 1)) = 12 , p((s, k), (s, k − 1)) = 1

2 for s > k

p((s, s), (s + 1, s + 1)) = 12 , p((s, s), (s, s − 1)) = 1

2

from which we can deduce the probability transitions of the Markov chain((Tn, Xn);n ≥ 1), with values in {(t, k) ∈ N

∗×Z | k ∈ (−t,−t+2, . . . , t−2, t)},

p((t, k), (t − 1, k + 1)) = 12 , p((t, k), (t + 1, k − 1)) = 1

2 if t > k (13.1)

p((t, t), (t + 1, t + 1)) = 12 , p((t, t), (t + 1, t − 1)) = 1

2

The transition probabilities are depicted in this picture.One checks then, by induction on n, that the conditional distribution of

Xn, knowing T1, . . . , Tn, is the uniform distribution on the set {−Tn,−Tn +2, . . . , Tn − 2, Tn}. Then it follows that (Tn;n ≥ 0) is a Markov chain withthe right transition probabilities.

112 P. Biane

13.2 A Markov Chain Associated with the Quantum

Bernoulli Random Walk

In section 5.2 we have seen that the quantum Bernoulli random walk givesrise in a natural way to two Markov chains, one being the classical Bernoullirandom walk, and the other being the spin process. These two processeswere obtained in the preceding section as coordinates of a certain two-dimensional Markov chain given by the transition probabilities (13.1). Wecan also consider a two-dimensional Markov chain having these two processesas marginals, by considering the Markov chain of the end of section 5.2.Recall that this Markov chain was obtained by restricting the generatorof the quantum Bernoulli random walk to the commutative subalgebraP(SU(2)) ⊂ A(SU(2)) generated by the center Z(SU(2)) and by a one para-meter subgroup. The spectrum of this algebra can be identified with the set

P = {(r, k) ∈ N × Z | k ∈ {−r,−r + 2, . . . , r − 2, r}}

Indeed this algebra is generated by the pair of commuting self-adjoint op-erators X, D in the sense that it consists in bounded functions of the pair(X, D). The joint spectrum of these operators can be computed from theexplicit description of the irreducible representations of SU(2), and coincideswith P . The probability transitions can be obtained by using the Clebsch-Gordan formula, or equivalently by the computation in the proof of Lemma3.3. One finds

p((r, k), (r + 1, k + 1)) =r + k + 2

2(r + 1)

p((r, k), (r + 1, k − 1)) =r − k + 2

2(r + 1)

p((r, k), (r − 1, k + 1)) =r − k

2(r + 1)

p((r, k), (r − 1, k − 1)) =r + k

2(r + 1).

These transition probabilities are on this pictureThus, although this Markov chain has the same one-dimensional marginal

as the one of the preceding section, they do not coincide. We will see thatin order to recover the transitions (13.1) we will have to introduce quantumgroups.

13.3 The Quantum Group SUq(2)

The Hopf algebra A(SU(2)) can be deformed by introducing a real para-meter q. The algebraic construction proceeds with the introduction of three

Introduction to Random Walks on Noncommutative Spaces 113

Fig. 9

generators t, e, f which are required to satisfy the relations

tet−1 = q2e, tft−1 = q−2f, ef − fe =t − t−1

q − q−1

and a coproduct which is given by

∆(t) = t ⊗ t, ∆(e) = e ⊗ t−1 + 1 ⊗ e, ∆(f) = f ⊗ 1 + t ⊗ f

Formally putting t = qh and letting q converge to 1 one finds in the limit thedefining relations for the envelopping algebra of the Lie algebra of SU(2), aswell as the coproduct.

One can prove that the irreducible finite dimensional representationsof this algebra are deformations of those of SU(2), indeed for each inte-ger r ≥ 0 there exists two representation in V +

r+1 and V −r+1, with bases

vr±k ; k ∈ {−r,−r + 2, . . . , r − 2, r}, given by

tvr±j = ±qjvr±

j

evr±j = ±

√

[

r − j

2

]

q

[

r + j + 2

2

]

q

vr±j+2

fvr±j =

√

[

r − j + 2

2

]

q

[

r + j

2

]

q

vr±j−2.

with [n]q = qn−q−n

q−q−1 . Using the coproduct one can define the tensor productof two representations, and this tensor product obeys the same rules as the

114 P. Biane

one for representations of SU(2) i.e. one has

V ǫ1r1+1 ⊗ V ǫ2

r2+1 =⊕

r=|r2−r1|,|r2−r1|+2,...,r1+r2

V ǫ1ǫ2r+1 .

We will now restrict our attention to representations of the kind V +l , and con-

sider the von Neumann-Hopf algebra A+(SUq(2)) = ⊕r≥0End[V +r+1], which

is isomorphic, as an algebra, to A(SU(2)), but whose coproduct is deformed.The subalgebra P(SUq(2)) generated by t and by the center remains un-changed in the deformation. We consider the tracial state 1

2Tr on the two-dimensional component, and consider the associated random walk. As in thecase of SU(2), the restriction of the associated Markov transtion operator tothe commutative algebra P(SUq(2)) defines a Markov chain on the spectrumof this algebra, whose transition probabilities can be computed, using thedeformed Clebsch Gordan formulas, as in Klimyk et Vilenkin [KV], formulas(6) et (9), §14.4.3, to give

p((r, k), (r + 1, k + 1)) = q(r−k)/2

[

r+k+22

]

q

[r + 1]q=

qr+1 − q−k−1

2(qr+1 − q−r−1)(13.2)

p((r, k), (r + 1, k − 1)) = q−(r+k)/2

[

r−k+22

]

q

2[r + 1]q=

q−k+1 − q−r−1

2(qr+1 − q−r−1)

p((r, k), (r − 1, k + 1)) = q−(r+k+2)/2

[

r−k2

]

q

2[r + 1]q=

q−k−1 − q−r−1

2(qr+1 − q−r−1)

p((r, k), (r − 1, k − 1)) = q(r−k+2)/2

[

r+k2

]

q

2[r + 1]q=

qr+1 − q−k+1

2(qr+1 − q−r−1)

Letting q tend to 0, one checks that (13.2) converges to (13.1), and thuswe get Pitman’s theorem as an outcome of the q → 0 limit of the quantumBernoulli random walk, see [B7] for details.

This observation is at the basis of a vast generalization of Pitman’s theo-rem, see e.g. [BBO].

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